#6454880

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437995
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×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3158 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)(11,15)(12,22)(13,17)(14,18)(16,25)(19,33)(20,29)(21,30)(23,28)(24,35)(26,38)(27,32)(31,37)(34,40)(36,39);(9,11)(10,15)(12,23,25,19)(13,26,27,20)(14,21)(16,28,22,33)(17,29,32,38)(18,30)(24,34)(31,39)(35,40)(36,37);(2,5)(6,8)(9,12,14,25)(10,16,18,22)(11,19,21,23)(13,24,27,37)(15,28,30,33)(17,31,32,35)(20,34,26,36)(29,39,38,40);(9,13)(10,17)(11,20)(12,24)(14,27)(15,29)(16,31)(18,32)(19,34)(21,26)(22,35)(23,36)(25,37)(28,39)(30,38)(33,40);(1,3,4,7)(2,6,5,8)(10,18)(15,30)(16,22)(17,32)(28,33)(29,38)(31,35)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362076);