#9065307

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

4380 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 8972503);