#510511

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

Polycyclic presentation:

2^9|a^2=efi,b^2=efi,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, 417707);