#507541

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=g,d^2=i,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, 414737);