Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(3,4)(17,18)(19,27)(20,21)(22,25)(23,26)(24,30)(28,31)(29,32);(1,2)(3,8)(4,7)(5,6)(10,13)(11,15)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,32)(26,27);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,25)(18,21,26,22)(19,28,30,32)(24,29,27,31);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,21,23,22)(18,25,26,20)(19,29,30,31)(24,32,27,28);(1,4,5,3)(2,7,6,8)(9,11,12,15)(10,14,13,16)(19,30)(24,27)(28,32)(29,31);(17,22,23,21)(18,25,26,20)(19,31,30,29)(24,32,27,28)Polycyclic presentation:
2^9|c^2=g,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012418);