#6453867

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437452
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×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3049 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^b=fi,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361063);