#508984

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
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

438 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,41)(30,42)(31,43)(32,40)(33,36)(34,35)(37,44)(38,45)(39,46)(47,48);(2,5)(6,8)(17,19,21,30,33,46,48,41,20,29,32,45,22,31,35,37)(18,23,25,38,36,43,47,44,24,27,40,42,26,39,34,28);(3,7)(6,8)(9,11,12,16)(10,13,14,15)(17,18)(19,28)(20,24)(21,34)(22,36)(23,37)(25,35)(26,33)(27,41)(29,44)(30,39)(31,38)(32,47)(40,48)(42,46)(43,45);(1,3,4,7)(2,6,5,8)(18,24)(23,27)(25,40)(26,36)(28,44)(34,47)(38,42)(39,43)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=ei,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, 416180);