#508863

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

559 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)(17,18,21,28,36,45,35,25,23,26,20,27,22,29,37,44)(19,24,32,41,48,38,47,43,34,39,31,40,33,42,30,46);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,20,31,36,48,37,30,23,34,21,32,22,33,35,47)(18,24,27,40,45,38,44,46,26,39,28,41,29,42,25,43);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(18,25)(19,30)(20,35)(21,37)(22,36)(24,38)(26,44)(27,29)(28,45)(31,48)(32,33)(34,47)(39,42)(43,46);(3,7)(6,8)(18,26)(24,39)(25,44)(27,28)(29,45)(38,42)(40,41)(43,46)

Polycyclic presentation:

2^9|a^2=g,b^2=f,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, 416059);