#510433

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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 ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

460 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,29,24)(20,31,33,25)(22,26,35,38)(28,39,45,43)(30,44,41,36)(32,40,48,46)(34,47,42,37);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,32)(21,29)(22,36)(23,27)(25,40)(26,41)(28,34)(30,38)(31,46)(33,48)(35,44)(37,39)(42,45)(43,47);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(17,20,21,33)(18,25,23,31)(19,28,29,45)(22,37,35,47)(24,39,27,43)(26,42,38,34)(30,46,41,40)(32,44,48,36);(1,5,6,4)(2,7,3,8)(10,14)(13,16)(19,29)(24,27)(28,45)(30,41)(32,48)(36,44)(39,43)(40,46)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417629);