#508865

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	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fh,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, 416061);