#9706334

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

4323 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)(3,8)(4,6)(5,7)(9,10,14,18)(11,19,21,15)(12,22,23,16)(13,17,20,24)(26,30)(27,33)(32,35)(34,39);(1,3,5,4)(2,6,7,8)(9,11,14,21)(10,15,18,19)(12,13,23,20)(16,17,22,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(9,12,14,23)(10,16,18,22)(11,20,21,13)(15,24,19,17)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,13)(14,23)(15,17)(18,22)(19,24)(20,21)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,12)(14,20)(15,16)(18,24)(19,22)(21,23)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40);(1,4,5,3)(2,6,7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,24)(22,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=hi,f^2=gh,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, 9613530);