#72281795

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346522
commutator, Frattini, cosocle	C₂×#131	C₂⁴
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3572 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:

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

Polycyclic presentation:

2^10|b^2=h,c^2=e,d^2=hj,f^2=i,i^2=j,b^a=bh,c^a=ce,c^b=cf,d^b=dh,d^c=defgi,e^b=eh,f^a=fh,f^b=fi,f^c=fhij,f^d=fi,f^e=fj,g^a=gh,g^b=ghj,g^c=ghj,g^d=gj,g^e=gj,g^f=gj,h^c=hj,i^b=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281944, #72307633, #72307980
maximal normal subgroup of	#72281944, #72307633, #72307980

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