#507147

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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 ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

486 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:

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

Polycyclic presentation:

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

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, 414343);