#6808133

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 6715329);