#499610

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

492 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,30,38)(20,24,33,41)(21,25,32,40)(27,35,44,47)(28,36,43,46)(29,37,34,42)(31,39,45,48);(2,5)(6,8)(17,19,22,30)(18,23,26,38)(20,27,33,44)(21,34,32,29)(24,35,41,47)(25,42,40,37)(28,45,43,31)(36,48,46,39);(9,11,12,16)(10,13,14,15)(19,27)(20,31)(21,32)(23,35)(24,39)(25,40)(28,29)(30,44)(33,45)(34,43)(36,37)(38,47)(41,48)(42,46);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406806);