#6408076

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6315272);