#6629115

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628872
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^b=ei,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, 6536311);