#72307364

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#8350
commutator	C₂×M₁₆	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂	C₂×#8350
prime radical	#10577624	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=h,c^2=h,d^2=fgj,e^2=i,f^2=i,g^2=j,i^2=j,b^a=bj,d^a=de,d^c=defij,e^a=ei,e^c=eh,f^a=fhij,f^c=fh,f^d=fj,f^e=fj,g^a=ghj,g^c=gh,g^d=gj,g^e=gj,h^d=hj,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307386, #72307601
maximal normal subgroup of	#72307386, #72307601

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