#10534345

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

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, 10441541);