#509040

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

459 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)(19,25)(20,34)(21,27)(22,28)(23,26)(24,31)(29,38)(30,39)(32,42)(33,41)(35,47)(36,44)(37,43)(40,45)(46,48);(1,2,4,5)(3,6,7,8)(17,19,29,45,20,30,37,36,21,31,22,32,23,33,35,46)(18,24,38,42,34,41,43,48,27,25,28,40,26,39,47,44);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,19,21,31)(18,25,27,24)(20,33,23,30)(22,36,29,46)(26,39,34,41)(28,44,38,48)(32,35,45,37)(40,43,42,47);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20,21,23)(18,26,27,34)(19,30,31,33)(22,35,29,37)(24,39,25,41)(28,43,38,47)(32,46,45,36)(40,48,42,44)

Polycyclic presentation:

2^9|b^2=eg,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, 416236);