#509062

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

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=e,d^2=fi,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, 416258);