#501902

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=e,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, 409098);