#6445811

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437999
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	C₂⁴	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2956 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=g,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,h^b=hi,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, 6353007);