#509004

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	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 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)(20,31)(21,25)(22,26)(23,27)(28,37)(29,38)(30,39)(32,46)(33,43)(34,40)(35,47)(36,42)(41,44)(45,48);(2,5)(6,8)(17,19,28,32,34,33,36,44,21,20,22,29,23,30,45,35)(18,24,37,46,40,43,42,41,25,31,26,38,27,39,48,47);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,20,21,19)(18,24,25,31)(22,35,28,44)(23,30,34,33)(26,41,37,47)(27,43,40,39)(29,36,32,45)(38,48,46,42);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(18,25)(24,31)(26,37)(27,40)(38,46)(39,43)(41,47)(42,48)

Polycyclic presentation:

2^9|b^2=gi,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, 416200);