#6629122

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628892
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	C₂⁴	C₂⁵
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4112 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)(9,10,14,15)(11,16,21,26)(12,17,23,27)(13,24,25,18)(19,31,33,28)(20,34,35,29)(22,36,37,30)(32,38,40,39);(1,3,4,8)(2,5,6,7)(10,15)(16,26)(17,27)(18,24)(28,31)(29,34)(30,36)(38,39);(9,11)(10,16)(12,19)(13,20)(14,21)(15,26)(17,28)(18,29)(22,32)(23,33)(24,34)(25,35)(27,31)(30,38)(36,39)(37,40);(9,12)(10,17)(11,19)(13,22)(14,23)(15,27)(16,28)(18,30)(20,32)(21,33)(24,36)(25,37)(26,31)(29,38)(34,39)(35,40);(9,13,14,25)(10,18,15,24)(11,20,21,35)(12,22,23,37)(16,29,26,34)(17,30,27,36)(19,32,33,40)(28,38,31,39)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,e^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,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, 6536318);