#6408025

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1339 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26,29,32,28,30,35,37)(27,31,34,38,33,36,39,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,17,19)(12,22,13,20)(16,24,18,23);(2,6)(3,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(26,30)(31,36)(32,37)(38,40);(2,6)(5,8)(10,17)(11,21)(16,18)(20,22)(26,30)(31,36)(32,37)(38,40);(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(25,27,29,34,28,33,35,39)(26,31,32,38,30,36,37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfgi,d^b=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6315221);