#6533146

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533141
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	#62901	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3295 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,g^b=gi,f^c=fi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440342);