#509020

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 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)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25)(19,20,28,24)(22,26,34,31)(23,27,35,40)(29,32,44,39)(30,33,43,38)(36,46,48,41)(37,42,47,45);(2,5)(6,8)(17,19,22,29,35,43,47,46,21,28,34,44,23,30,37,41)(18,24,26,39,40,33,45,36,25,20,31,32,27,38,42,48);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,20,21,24)(18,19,25,28)(22,36,34,48)(23,38,35,33)(26,41,31,46)(27,43,40,30)(29,45,44,42)(32,47,39,37);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(18,25)(20,24)(26,31)(27,40)(32,39)(33,38)(36,48)(42,45)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416216);