#510453

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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

450 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)(20,25,32,38)(22,26,34,37)(28,43,45,39)(30,44,41,35)(31,46,47,40)(33,36,42,48);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,31)(21,29)(22,35)(23,27)(25,40)(26,41)(28,33)(30,37)(32,47)(34,44)(36,39)(38,46)(42,45)(43,48);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(17,20,21,32)(18,25,23,38)(19,28,29,45)(22,36,34,48)(24,39,27,43)(26,42,37,33)(30,46,41,40)(31,44,47,35);(1,5,6,4)(2,7,3,8)(10,14)(13,16)(19,29)(24,27)(28,45)(30,41)(31,47)(35,44)(39,43)(40,46)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417649);