#6500022

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500019
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 ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3313 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)(11,15)(12,22,24,16)(13,17)(14,18)(19,26,31,34)(20,27)(21,28)(23,35,36,29)(25,30)(32,37,39,40)(33,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,26)(17,28)(18,27)(22,34)(23,32)(24,31)(25,33)(29,37)(30,38)(35,40)(36,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,23)(14,24)(15,26)(17,29)(18,22)(20,31)(21,32)(25,36)(27,34)(28,37)(30,35)(33,39)(38,40);(3,8)(5,7)(11,20)(15,27)(19,31)(21,33)(26,34)(28,38)(32,39)(37,40);(2,6)(5,7)(9,13,14,25)(10,17,18,30)(11,21,20,33)(12,23,24,36)(15,28,27,38)(16,29,22,35)(19,32,31,39)(26,37,34,40)

Polycyclic presentation:

2^9|a^2=f,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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303332, #72303431, #72303511
maximal normal subgroup of	#72303332, #72303431, #72303511

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407218);