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#72285721

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151462
commutator, Frattini, cosocle#1781C23
prime radicalC23×D4C2×D4
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

423 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 37 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)(4,18)(5,10,25,37)(6,26,28,11)(7,12)(8,17)(9,20)(14,42,44,61)(15,35,46,30)(16,31,47,38)(19,51,53,48)(21,56)(22,50)(23,34)(24,40)(27,36)(29,60,41,43)(32,52,63,54)(33,55)(39,57)(45,62)(49,58)(59,64);(1,3)(2,8,12,13)(4,19)(5,15,25,46)(6,16)(7,17)(9,32,34,63)(10,35)(11,38,26,31)(14,21)(18,48)(20,54,23,52)(22,53)(24,59,27,45)(28,47)(29,39,41,33)(30,37)(36,64)(40,62)(42,56)(43,55,60,57)(44,58)(49,61)(50,51);(1,4)(2,9,12,34)(3,14,17,44)(5,23)(6,21,28,58)(7,22)(8,29)(10,18,37,50)(11,39)(13,41)(15,43,46,60)(16,19)(20,25)(24,55,27,57)(26,33)(30,42)(31,63,38,32)(35,61)(36,49)(40,56)(45,54)(47,53)(48,62,51,64)(52,59)

Polycyclic presentation:

2^10|a^2=fgh,b^2=g,c^2=gh,d^2=gh,e^2=g,b^a=bh,c^a=cd,c^b=ce,d^a=dg,d^b=df,d^c=dghj,e^a=efhj,e^b=eg,e^c=egij,e^d=ehj,f^a=fj,f^b=fj,f^c=fh,f^d=fij,f^e=fi,g^a=gij,g^d=gj,h^a=hi,h^b=hij,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285757
maximal normal subgroup of#72285757

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