#6811626

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=hi,d^2=f,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718822);