#72264463

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#410
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂×#117	C₂⁴
socle	C₂²	C₂×#1820
prime radical	C₂⁵	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

1646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(3,6)(7,8)(9,10,13,18,14,16,26,32)(11,19,21,36,22,35,25,34)(12,17,24,33,20,28,38,40)(15,27,31,23,30,39,29,37);(1,2,6,8,4,7,3,5)(9,11,20,37)(10,15,28,35)(12,23,14,22)(13,25,38,39)(16,30,17,19)(18,29,40,36)(21,24,27,26)(31,33,34,32);(2,7)(5,8)(10,16)(11,21,22,25)(15,29,30,31)(17,28)(18,32)(19,34,35,36)(23,39,37,27)(33,40);(2,7)(5,8)(11,22)(15,30)(19,35)(21,25)(23,37)(27,39)(29,31)(34,36)

Polycyclic presentation:

2^10|a^2=g,b^2=ef,c^2=hj,e^2=ij,g^2=j,b^a=be,c^a=cj,c^b=cgij,d^b=dij,e^a=ei,e^b=eh,f^a=fhj,f^b=fh,f^e=fj,g^b=gj,h^b=hj

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264470
maximal normal subgroup of	#72264470

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