#6440466

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437632
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	#62859	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4183 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,27)(20,28)(21,26)(22,30)(23,29)(24,31)(25,32)(33,38)(34,37)(35,40)(36,39);(1,3,4,7)(2,5,6,8)(15,26)(16,29)(28,31)(30,39);(2,6)(5,8)(9,11,14,21)(10,15,18,26)(12,19,23,34)(13,24,25,20)(16,27,29,37)(17,31,32,28)(22,35,36,33)(30,40,39,38);(9,12)(10,16)(11,19)(13,22)(14,23)(15,27)(17,30)(18,29)(20,33)(21,34)(24,35)(25,36)(26,37)(28,38)(31,40)(32,39);(9,13,14,25)(10,17,18,32)(11,20,21,24)(12,22,23,36)(15,28,26,31)(16,30,29,39)(19,33,34,35)(27,38,37,40)

Polycyclic presentation:

2^9|b^2=g,c^2=i,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,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, 6347662);