#72280758

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10918
commutator	C₂×#41	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#124038
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

3887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=h,c^2=dej,d^2=j,e^2=j,b^a=bh,c^a=cd,c^b=cg,d^b=dj,e^a=ej,e^b=ej,f^c=fj,g^a=gi,g^b=gi,h^c=hi,h^g=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280775, #72281237, #72281807
maximal normal subgroup of	#72280775, #72281237, #72281807

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