#507557

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

544 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)(3,7)(4,5)(6,8)(17,18,22,27)(19,23,30,38)(20,24,33,40)(21,34,32,26)(25,41,39,31)(28,35,44,47)(29,45,43,37)(36,48,46,42);(2,5)(6,7)(9,10,12,14)(11,15,16,13)(17,19,22,30)(18,23,27,38)(20,28,33,44)(21,29,32,43)(24,35,40,47)(25,36,39,46)(26,37,34,45)(31,42,41,48);(9,11,12,16)(10,13,14,15)(18,24)(20,31)(21,32)(23,35)(25,26)(27,40)(28,42)(29,43)(33,41)(34,39)(36,37)(38,47)(44,48)(45,46);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414753);