#499330

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

784 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)(10,17)(11,23)(12,13)(14,30)(15,29)(18,36)(19,34)(20,35)(21,22)(25,27)(26,28)(31,32)(37,40)(38,39);(5,6)(7,8)(9,10,13,20,15,22,30,34,16,19,14,21,29,35,12,17)(11,24,23,40,27,38,31,18,28,33,26,36,32,39,25,37);(2,4)(6,8)(9,11,16,28)(10,18,19,37)(12,23,30,26)(13,25,14,31)(15,32,29,27)(17,33,34,24)(20,38,21,39)(22,40,35,36);(10,19)(17,34)(18,37)(20,21)(22,35)(24,33)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^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,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, 406526);