#509039

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

463 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:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=i,d^2=f,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, 416235);