#510510

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

635 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ef,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417706);