#72307312

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346546
commutator, Frattini, cosocle	C₂×#131	C₂⁴
prime radical	#10584533	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3361 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:

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

Polycyclic presentation:

2^10|a^2=hj,b^2=hj,c^2=fij,d^2=h,e^2=j,f^2=ij,g^2=j,i^2=j,b^a=bh,c^a=ce,c^b=cefi,d^a=dh,d^c=degj,e^d=eh,f^a=fj,f^b=fi,g^a=gh,g^b=ghj,g^c=gj,g^d=gh,g^e=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307386, #72307865, #72308079
maximal normal subgroup of	#72307386, #72307865, #72308079

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