#6572294

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	#62905	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3073 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,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, 6479490);