#407865

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

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

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=e,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315061);