#6629194

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 ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4103 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,16,22,30)(12,25,27,34)(13,19,14,18)(17,28,33,26)(21,35,36,40)(23,32,24,31)(29,38,39,37);(2,5)(6,8)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,29)(18,31)(19,32)(20,30)(25,35)(26,37)(27,36)(28,38)(33,39)(34,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,28)(14,26)(15,27)(16,29)(18,34)(19,25)(20,33)(22,36)(23,38)(24,37)(30,39)(31,40)(32,35);(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,29)(18,25)(19,34)(20,33)(22,36)(23,37)(24,38)(30,39)(31,35)(32,40);(11,22)(16,30)(21,36)(23,24)(29,39)(31,32)(35,40)(37,38)

Polycyclic presentation:

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

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, 6536390);