#507801

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=gi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414997);