#9683085

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

4355 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9590281);