Finite Group Databasebeta

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#72283521

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#149761
commutatorC23×D4C22×C4
Frattini, cosocle#1781C23
prime radicalC22×H(5,𝔽2)C2×C4
LCSG ▷ C23×D4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

991 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 991 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,13,18,35)(4,19)(5,10,26,38)(6,11)(7,12)(8,16)(9,23)(14,37,45,28)(15,41,20,53)(17,42)(21,36,25,33)(22,34)(24,31)(27,40)(29,49)(30,43,57,50)(32,47)(39,54)(44,55,46,58)(48,64)(51,61)(52,60)(56,62,59,63);(1,3,14,43)(2,8,28,42)(4,20,44,62,23,15,46,63)(5,24,27,52,26,16,6,17)(7,18,45,50)(9,32,58,64,19,29,55,61)(10,35,11,30,38,13,40,57)(12,31,37,60)(21,49,22,48)(25,47,54,51)(33,41,39,59)(34,56,36,53);(1,4)(2,9)(3,15)(5,25)(6,22)(7,23)(8,29)(10,36)(11,39)(12,19)(13,41)(14,44)(16,49)(17,51)(18,20)(21,26)(24,47)(27,54)(28,55)(30,56)(31,32)(33,38)(34,40)(35,53)(37,58)(42,61)(43,62)(45,46)(48,52)(50,63)(57,59)(60,64)

Polycyclic presentation:

2^10|a^2=fgij,b^2=deg,e^2=g,b^a=bd,c^a=cij,c^b=cfghi,d^a=dhij,d^b=df,d^c=dj,e^a=efhj,e^b=ef,e^c=ehj,e^d=ehj,f^a=fij,f^b=fh,f^c=fj,f^e=fi,g^a=gij,g^b=gi,g^c=gj,g^d=gj,h^a=hj,h^b=hi,h^e=hj,i^b=ij

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283564, #72283861, #72284081, …
maximal normal subgroup of#72283564, #72283861, #72284081, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 2, F.4*F.5*F.7);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);