#72281111

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, socle	C₂	#392585
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3584	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #60058 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5950 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)(3,14)(4,20)(5,12)(6,11)(7,13)(8,15)(9,25)(10,27)(16,48)(17,32)(18,45)(19,34)(21,57)(22,55)(23,41)(24,40)(26,56)(28,37)(29,62)(30,43)(31,53)(33,47)(35,50)(36,49)(38,61)(39,60)(42,51)(44,52)(46,63)(54,58)(59,64);(1,3,7,15)(2,8,13,14)(4,21,24,58)(5,17,27,19)(6,29,30,51)(9,35,38,64)(10,32,12,34)(11,42,43,62)(16,49,46,26)(18,23,47,28)(20,54,40,57)(22,53,60,52)(25,59,61,50)(31,55,44,39)(33,41,45,37)(36,48,56,63);(3,15)(4,22,24,60)(5,27)(6,23)(8,14)(9,36,38,56)(10,12)(11,41)(16,50,46,59)(18,29)(20,55,40,39)(21,52,58,53)(25,49,61,26)(28,30)(31,57,44,54)(33,42)(35,63,64,48)(37,43)(45,62)(47,51);(1,4,23,22,5,25,6,26,7,24,28,60,27,61,30,49)(2,9,37,36,12,40,43,55,13,38,41,56,10,20,11,39)(3,16,51,59,19,53,18,21,15,46,29,50,17,52,47,58)(8,31,42,54,32,63,45,64,14,44,62,57,34,48,33,35)

Polycyclic presentation:

2^10|b^2=j,c^2=e,d^2=fghi,f^2=j,g^2=ij,h^2=j,i^2=j,b^a=bj,c^b=cj,d^a=df,d^b=dg,d^c=dfghi,e^d=ej,f^a=fj,f^c=fj,g^b=gi,g^c=gij,h^c=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281386
maximal normal subgroup of	#72281386

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