#6573417

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

3190 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)(12,22)(13,25,26,17)(16,24)(19,27)(20,33,34,28)(23,30)(31,39)(32,35)(36,37)(38,40);(1,2,6,5)(3,8,4,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,27)(17,28)(18,29)(22,35)(23,31)(24,32)(25,33)(26,34)(30,38)(36,39)(37,40);(1,3,6,4)(2,7,5,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,30)(18,22)(20,31)(21,32)(25,36)(26,37)(28,38)(29,35)(33,39)(34,40);(1,4,6,3)(2,7,5,8)(9,13,14,26)(10,17,18,25)(11,20,21,34)(12,23,24,37)(15,28,29,33)(16,30,22,36)(19,31,32,40)(27,38,35,39);(1,5,6,2)(3,8,4,7)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=h,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,g^a=gi,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, 6480613);