Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,14,18)(11,15,21,23)(12,16,22,20)(13,17,24,19)(25,26)(27,36)(28,33)(29,38)(30,34)(31,35)(32,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,17)(13,20)(14,21)(16,24)(18,23)(19,22)(25,27)(26,32)(28,30)(29,34)(31,37)(33,39)(35,36)(38,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,15)(14,22)(17,21)(18,20)(23,24)(25,28,31,38)(26,33,35,29)(27,30,37,40)(32,39,36,34);(1,4,5,6)(2,3,7,8)(9,12)(10,16)(11,17)(13,23)(14,22)(15,24)(18,20)(19,21)(25,29)(26,28)(27,34)(30,32)(31,33)(35,38)(36,40)(37,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,19)(13,16)(14,21)(17,22)(18,23)(20,24)(25,26)(27,32)(28,33)(29,38)(30,39)(31,35)(34,40)(36,37);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,20)(12,15)(14,24)(16,21)(18,19)(22,23)(25,30,31,40)(26,34,35,39)(27,28,37,38)(29,36,33,32)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=h,f^2=g,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=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9121969);