#507268

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414464);