Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11307 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #1443 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1502 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)(5,9,14,23)(6,16)(7,20)(8,11)(10,18)(12,21)(13,22,27,34)(15,29,28,24)(17,30,32,25)(19,26)(31,33);(3,5)(4,9,11,23)(6,17,18,32)(7,15)(8,14)(10,25)(12,24,20,29)(13,19,27,33)(16,30)(21,28)(22,31)(26,34);(3,6)(4,10,11,16)(5,13,14,27)(7,19)(8,18)(9,22)(12,26,20,31)(15,17,28,32)(21,33)(23,34)(24,25)(29,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265541);