#509000

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	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

541 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:

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

Polycyclic presentation:

2^9|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, 416196);