#6837138

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806551
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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1401 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=ghi,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=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, 6744334);