#509003

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

562 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)(19,24)(20,25)(21,27)(22,28)(23,26)(29,37)(30,38)(31,39)(32,40)(33,41)(34,43)(35,42)(36,44)(45,47)(46,48);(2,5)(6,8)(17,19,29,32,35,33,36,45,23,20,21,30,22,31,46,34)(18,24,37,40,42,41,44,47,26,25,27,38,28,39,48,43);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,20,23,19)(18,25,26,24)(21,34,29,45)(22,31,35,33)(27,43,37,47)(28,39,42,41)(30,36,32,46)(38,44,40,48);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(18,26)(24,25)(27,37)(28,42)(38,40)(39,41)(43,47)(44,48)

Polycyclic presentation:

2^9|b^2=gi,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, 416199);