#6453857

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

3116 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)(11,20)(12,18)(13,19)(14,26)(15,17)(16,22)(21,36)(23,30)(24,34)(25,32)(27,31)(28,35)(29,37)(33,38)(39,40);(9,11)(10,16)(12,21)(13,22)(14,27)(15,23)(17,31)(18,33)(19,30)(20,26)(24,37)(25,39)(28,38)(29,32)(34,36)(35,40);(2,5)(6,8)(10,17)(13,14)(18,32)(20,30)(22,27)(24,28)(36,40)(37,38);(9,12,15,25)(10,18,17,32)(11,21,23,39)(13,24,14,28)(16,29,31,33)(19,34,26,35)(20,36,30,40)(22,37,27,38);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,19,17,26)(11,22,23,27)(12,24,25,28)(16,30,31,20)(18,34,32,35)(21,37,39,38)(29,40,33,36)

Polycyclic presentation:

2^9|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=dh,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, 6361053);