#9229749

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

4540 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)(11,21)(12,18)(13,14)(15,19)(16,20)(17,24)(22,23)(25,26,30,33)(27,34,28,31)(29,37,38,32)(35,39,36,40);(1,2)(3,4)(5,7)(6,8)(9,11,16,24)(10,17,20,21)(12,22,13,19)(14,23,18,15)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(1,3)(2,6)(4,7)(5,8)(9,12,16,13)(10,18,20,14)(11,22,24,19)(15,21,23,17)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,2)(3,6)(4,8)(5,7)(9,13,16,12)(10,18,20,14)(11,19,24,22)(15,21,23,17)(25,26)(27,31)(28,34)(29,37)(30,33)(32,38)(35,40)(36,39);(1,4,5,6)(2,3,7,8)(9,14,16,18)(10,13,20,12)(11,23,24,15)(17,19,21,22)(25,28,30,27)(26,31,33,34)(29,35,38,36)(32,40,37,39);(1,3)(2,4)(5,8)(6,7)(9,15,16,23)(10,19,20,22)(11,14,24,18)(12,21,13,17)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=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, 9136945);