#509032

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

482 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:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=i,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,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, 416228);