#499328

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

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^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, 406524);