#6533258

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

3205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440454);