#6834484

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei

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, 6741680);