#6533540

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3331 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)(11,20)(12,17)(13,19)(14,26)(15,18)(16,22)(21,36)(23,31)(24,33)(25,32)(27,30)(28,34)(29,38)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,22)(14,27)(15,23)(17,29)(18,30)(19,31)(20,26)(24,37)(25,39)(28,38)(32,40)(33,36)(34,35);(2,5)(6,8)(9,12)(10,17)(11,21)(13,24)(14,28)(15,25)(16,29)(18,32)(19,33)(20,35)(22,37)(23,39)(26,34)(27,38)(30,40)(31,36);(1,3)(2,6)(4,7)(5,8)(10,18)(16,30)(17,32)(19,26)(20,31)(29,40)(33,34)(35,36);(3,7)(6,8)(9,13,15,14)(10,19,18,26)(11,22,23,27)(12,24,25,28)(16,31,30,20)(17,33,32,34)(21,37,39,38)(29,36,40,35)

Polycyclic presentation:

2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,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	#72317675, #72319333
maximal normal subgroup of	#72317675, #72319333

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440736);