#510161

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

478 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=f,c^2=e,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, 417357);