#507359

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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

454 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fg,c^2=i,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,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, 414555);