#6617487

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573715
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
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524683);