#6453868

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

Polycyclic presentation:

2^9|b^2=i,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, 6361064);