#6811495

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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

1993 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)(12,26)(13,19)(14,30)(15,20)(16,21)(18,23)(22,36)(24,34)(25,35)(27,39)(28,37)(29,40)(31,38)(32,33);(1,3,6,2)(4,8,5,7)(9,11)(10,17)(12,27)(13,28)(14,24)(15,32)(16,25)(18,36)(19,37)(20,33)(21,35)(22,23)(26,39)(29,31)(30,34)(38,40);(9,12,15,23,16,28,14,31)(10,18,20,37,21,38,30,26)(11,22,24,27,25,29,32,13)(17,19,34,36,35,39,33,40);(1,4,6,5)(2,7,3,8);(1,5,6,4)(2,7,3,8)(9,13,15,22,16,27,14,29)(10,19,20,36,21,39,30,40)(11,23,24,28,25,31,32,12)(17,18,34,37,35,38,33,26)

Polycyclic presentation:

2^9|b^2=f,c^2=h,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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718691);