Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11313 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #1443 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,14,17,10)(6,18)(7,22)(8,13)(9,12)(11,20)(15,32,34,28)(16,26,23,27)(19,25,33,31)(21,30)(24,29);(3,5,7,16,9,17,8,23)(4,10,22,27,12,14,13,26)(6,19,20,15,21,33,24,34)(11,28,30,25,29,32,18,31);(3,6)(4,11,12,29)(5,15)(7,20)(8,24)(9,21)(10,25,14,31)(13,18,22,30)(16,33)(17,34)(19,23)(26,28,27,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265546);