#509063

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

398 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=ei,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416259);