#512504

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,24,28)(19,29,35,43)(20,36,39,40)(21,38,41,37)(22,27,23,26)(25,34,44,32)(30,31,42,33)(45,48,46,47);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,32,24,35,23,34)(18,25,26,29,28,44,27,43)(20,30,37,46,39,42,38,45)(21,33,36,47,41,31,40,48);(1,3,5,4)(2,7,6,8)(17,20,36,28,23,38,21,26,24,39,40,18,22,37,41,27)(19,30,47,44,34,45,33,29,35,42,48,25,32,46,31,43);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,22,36,24,41,23,40)(18,20,26,37,28,39,27,38)(19,31,32,48,35,33,34,47)(25,42,29,45,44,30,43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ef,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,g^a=gi,h^a=hi,d^b=di

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, 419700);