#507781

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

566 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,22,37)(20,30,32,47)(21,33,36,46)(25,40,42,31)(26,29,44,35)(28,45,38,43)(34,39,48,41);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,22)(18,24,27,37)(20,28,32,38)(21,29,36,35)(25,39,42,41)(26,33,44,46)(30,45,47,43)(31,34,40,48);(11,16)(13,15)(17,20,23,32)(18,25,27,42)(19,28,22,38)(21,34,36,48)(24,39,37,41)(26,43,44,45)(29,40,35,31)(30,46,47,33);(1,3,4,8)(2,6,5,7)(10,14)(13,15)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=eg,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, 414977);