#9706346

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

4396 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gh,e^2=ghi,f^2=h,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, 9613542);