#6811623

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

1982 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)(11,16)(12,14)(13,17)(15,19)(18,30)(20,24)(21,31)(22,25)(23,32)(26,29)(27,33)(28,34)(35,40)(36,37)(38,39);(1,3,6,2)(4,8,5,7)(9,11)(10,16)(12,25)(13,27)(14,22)(15,24)(17,33)(18,32)(19,20)(21,29)(23,30)(26,31)(28,38)(34,39)(35,36)(37,40);(9,12,14,19,15,18,30,10)(11,20,23,25,24,16,22,32)(13,26,29,34,28,35,40,17)(21,36,27,39,37,31,38,33);(1,4,6,5)(2,7,3,8)(13,28)(17,34)(21,37)(26,35)(27,38)(29,40)(31,36)(33,39);(1,5,6,4)(2,7,3,8)(9,13,14,29,15,28,30,40)(10,17,12,26,19,34,18,35)(11,21,23,27,24,37,22,38)(16,31,32,33,20,36,25,39)

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,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, 6718819);