#507409

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egi,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414605);