Finite Group Databasebeta

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#72285336

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#142537
commutator, Frattini, cosocle#2309C23
prime radical#23445C22
LCSG ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G
LECSG ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1051 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 1051 conjugacy classes of subgroups — too many to draw. Only its 59 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=fhij,c^2=ghj,f^2=hi,b^a=bf,c^a=ce,c^b=cd,d^a=dh,d^b=dg,d^c=dij,e^b=egh,e^c=ei,e^d=ej,f^a=fhi,f^c=fgj,f^d=fi,f^e=fij,g^a=gi,g^b=gi,g^c=gj,g^f=gj,h^b=hj,h^c=hj,i^b=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285412, #72285854, #72285911
maximal normal subgroup of#72285412, #72285854, #72285911

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.10);;
gap> SetPower(coll, 2, F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);