#501881

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	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

524 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)(19,28)(24,39)(29,44)(30,38)(31,45)(35,48)(40,41)(43,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,29)(20,25)(21,26)(22,34)(23,37)(24,31)(27,36)(28,44)(30,47)(32,42)(33,46)(35,40)(38,43)(39,45)(41,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,19,21,28)(18,24,26,39)(20,30,23,38)(22,35,33,48)(25,40,37,41)(27,43,42,47)(29,46,44,34)(31,32,45,36);(2,5)(6,8)(17,20,21,23)(18,25,26,37)(19,30,28,38)(22,32,33,36)(24,40,39,41)(27,34,42,46)(29,47,44,43)(31,35,45,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=ei,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409077);