Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C4×#1796 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6)(7,15)(8,19,21,12)(9,22)(10,14)(11,17)(13,24)(16,30,32,29)(18,28)(20,27,36,34)(23,26)(25,31,33,35);(5,7,10,18)(6,11,14,26)(8,16,21,32)(9,23,24,17)(12,25,19,33)(13,28,22,15)(20,35,36,31)(27,30,34,29);(5,8)(6,12)(7,16)(9,20)(10,21)(11,25)(13,27)(14,19)(15,29)(17,31)(18,32)(22,34)(23,35)(24,36)(26,33)(28,30)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262017);