Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C4×#1902 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1789 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)(7,16)(8,20,23,30)(9,24)(10,14)(11,15)(12,18)(13,22,29,21)(17,32,34,26)(19,27)(25,28)(31,33,36,35);(5,7,9,18,11,19,10,25)(6,12,14,16,15,28,24,27)(8,17,22,35,23,34,21,33)(13,26,30,31,29,32,20,36);(5,8)(6,13)(7,17)(9,22)(10,21)(11,23)(12,26)(14,30)(15,29)(16,31)(18,35)(19,34)(20,24)(25,33)(27,36)(28,32)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261447);