#507517

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=eg,c^2=ef,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414713);