#6500026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500015
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 ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3240 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)(11,19,22,15)(12,16)(13,17)(14,18)(20,33,36,28)(21,34,38,29)(23,25)(24,30)(26,31)(27,32)(35,40,37,39);(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,23)(13,17)(14,18)(16,25)(19,22)(20,33)(21,29)(24,31)(26,30)(27,32)(28,36)(34,38)(35,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,28)(17,29)(18,19)(23,33)(24,35)(25,36)(26,37)(27,38)(30,39)(31,40)(32,34);(3,8)(5,7)(9,12,14,25)(10,16,18,23)(11,20,22,36)(13,26,27,24)(15,28,19,33)(17,31,32,30)(21,37,38,35)(29,40,34,39);(2,6)(5,7)(9,13,14,27)(10,17,18,32)(11,21,22,38)(12,24,25,26)(15,29,19,34)(16,30,23,31)(20,35,36,37)(28,39,33,40)

Polycyclic presentation:

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

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, 6407222);