#512488

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,28,41)(20,32,34,47)(21,35,33,25)(23,36,37,30)(24,39,40,29)(31,46,44,43)(38,45,42,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(19,28)(23,37)(27,41)(30,36)(31,44)(38,42)(43,46)(45,48);(17,19,29,45,21,31,34,36,22,28,39,48,33,44,20,30)(18,23,32,43,25,38,40,41,26,37,47,46,35,42,24,27);(1,4,5,3)(2,6,7,8)(10,14)(13,16)

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,c^b=ci

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, 419684);