#6452277

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	#23445	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3344 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,26)(14,18)(17,28)(19,22)(20,29)(21,32)(23,25)(24,36)(27,30)(31,38)(33,35)(34,40)(37,39);(9,10)(11,19,22,15)(12,23,25,16)(13,17)(14,18)(20,29)(21,32,27,30)(24,36,38,31)(26,28)(33,35)(34,39)(37,40);(1,3,5,4)(2,6,7,8)(9,11,14,22)(10,15,18,19)(12,20,25,35)(13,27,28,21)(16,29,23,33)(17,32,26,30)(24,37,38,34)(31,40,36,39);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,20)(13,24)(14,25)(15,29)(17,31)(18,23)(19,33)(21,34)(22,35)(26,36)(27,37)(28,38)(30,39)(32,40);(9,13,14,28)(10,17,18,26)(11,21,22,27)(12,24,25,38)(15,30,19,32)(16,31,23,36)(20,34,35,37)(29,39,33,40)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=g,e^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6359473);