#9064168

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

4387 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8971364);