#72320009

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#44871
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#777670
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5139 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gi,c^2=g,e^2=i,f^2=j,h^2=j,b^a=bh,c^a=cj,c^b=cj,d^a=df,d^b=dfhj,d^c=dfj,e^a=ehj,e^b=eg,e^c=egj,e^d=eghi,f^d=fj,f^e=fj,g^d=gj,h^a=hj,h^d=hj,i^a=ij,i^d=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320148
maximal normal subgroup of	#72320148

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