#6867658

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1677 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6774854);