#507542

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
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

537 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,25)(19,28,29,23)(21,24,32,40)(22,35,33,27)(26,42,41,34)(30,36,45,39)(31,38,46,44)(37,48,43,47);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,23)(20,29)(21,30)(22,31)(24,39)(25,28)(26,37)(27,38)(32,45)(33,46)(34,48)(35,44)(36,40)(41,43)(42,47);(9,11,12,15)(10,14,13,16)(18,24)(19,29)(21,34)(22,33)(23,36)(25,40)(26,27)(28,39)(30,47)(32,42)(35,41)(37,44)(38,43)(45,48);(2,6)(5,7)(10,13)(14,16)(18,25)(19,29)(24,40)(26,41)(27,35)(30,45)(31,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=gi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,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, 414738);