#499334

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=i,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^b=di,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, 406530);