#507449

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
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

524 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)(9,10,12,14)(11,13,15,16)(19,27)(20,32)(22,33)(23,35)(25,40)(26,39)(28,41)(29,34)(30,43)(31,46)(36,38)(37,48)(42,44)(45,47);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,18,21,24)(19,28,30,23)(20,25,34,38)(22,26,33,39)(27,41,43,35)(29,36,32,40)(31,45,42,37)(44,48,46,47);(11,15)(13,16)(17,19,29,46,22,31,20,27,21,30,32,44,33,42,34,43)(18,23,36,48,26,37,25,35,24,28,40,47,39,45,38,41);(2,6)(5,7)(18,24)(23,28)(25,38)(26,39)(35,41)(36,40)(37,45)(47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=eg,c^2=fhi,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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414645);