#6808125

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

1748 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=f,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, 6715321);