#6629111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628867
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536307);