#510425

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

456 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)(18,23)(19,27)(25,39)(26,36)(29,44)(30,37)(33,45)(34,48);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,18)(19,28)(20,23)(21,33)(22,36)(24,29)(25,35)(26,32)(27,43)(30,47)(31,45)(34,42)(37,41)(38,44)(39,46)(40,48);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(17,19,20,27)(18,24,23,38)(21,34,31,48)(22,37,32,30)(25,41,39,47)(26,42,36,40)(28,45,43,33)(29,46,44,35);(1,5,6,4)(2,7,3,8)(10,14)(13,16)(18,23)(24,38)(25,39)(26,36)(28,43)(33,45)(40,42)(41,47)

Polycyclic presentation:

2^9|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, 417621);