#6533342

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
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3422 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)(11,16)(12,17)(13,27)(14,18)(15,19)(20,33)(21,37)(22,29)(23,30)(24,38)(25,31)(26,32)(28,36)(34,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,28)(18,29)(19,30)(24,34)(25,35)(26,36)(27,37)(31,39)(32,33)(38,40);(2,5)(6,8)(9,12)(10,17)(11,20)(13,24)(14,25)(15,26)(16,28)(18,31)(19,32)(21,34)(22,35)(23,36)(27,38)(29,39)(30,33)(37,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,14)(10,18,19,27)(11,21,23,22)(12,24,26,25)(16,29,30,37)(17,31,32,38)(20,34,36,35)(28,39,33,40);(3,7)(6,8)(9,14,15,13)(10,18,19,27)(11,22,23,21)(12,25,26,24)(16,29,30,37)(17,31,32,38)(20,35,36,34)(28,39,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303332, #72303523, #72303533
maximal normal subgroup of	#72303332, #72303523, #72303533

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440538);