#9065263

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

4600 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=g,f^2=gh,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, 8972459);