#508867

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

563 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)(17,18,22,29,20,38,39,46,21,28,23,30,24,27,40,26)(19,31,35,44,33,42,32,45,34,25,36,43,37,41,47,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,35,20,33,39,32,21,34,23,36,24,37,40,47)(18,25,29,43,38,41,46,48,28,31,30,44,27,42,26,45);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(18,26)(19,32)(20,24)(22,40)(23,39)(25,41)(27,29)(28,46)(30,38)(31,42)(33,35)(34,47)(36,37)(45,48);(3,7)(6,8)(17,20,21,24)(18,27,28,38)(19,33,34,37)(22,39,23,40)(25,42,31,41)(26,30,46,29)(32,36,47,35)(43,45,44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=i,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, 416063);