#6808126

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

1770 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)(9,10,16,19)(11,17,23,20)(12,14,26,15)(13,27,31,18)(21,36,37,33)(22,34,32,24)(25,30,39,29)(28,38,40,35);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,24)(13,28)(14,22)(15,32)(16,23)(18,35)(19,20)(21,30)(25,33)(26,34)(27,38)(29,37)(31,40)(36,39);(9,12,15,19,16,26,14,10)(11,20,22,24,23,17,32,34)(13,25,30,27,31,39,29,18)(21,36,28,38,37,33,40,35);(1,3,4,7)(2,6,5,8);(2,5)(6,8)(9,13,14,29,16,31,15,30)(10,18,26,39,19,27,12,25)(11,21,32,40,23,37,22,28)(17,33,24,38,20,36,34,35)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=f,e^2=hi,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, 6715322);