#6810891

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
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

1655 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=f,e^2=fhi,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, 6718087);