#499327

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=i,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,d^c=di,e^c=eh,f^c=fhi,g^c=ghi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406523);