#501900

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 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)(17,18,22,26)(19,23,28,36)(20,24,33,41)(21,25,32,40)(27,35,44,47)(29,37,34,42)(30,38,43,46)(31,39,45,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(19,27)(20,30)(21,32)(23,35)(24,38)(25,40)(28,44)(29,45)(31,34)(33,43)(36,47)(37,48)(39,42)(41,46);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,19,22,28)(18,23,26,36)(20,31,33,45)(21,34,32,29)(24,39,41,48)(25,42,40,37)(27,43,44,30)(35,46,47,38);(2,5)(6,8)(19,28)(23,36)(27,44)(29,34)(31,45)(35,47)(37,42)(39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,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, 409096);