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×#11314 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| 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);(3,4,8,13,9,12,7,14)(5,15,17,26,19,10,18,27)(6,20,23,29,24,30,22,11)(16,28,21,31,32,34,33,25);(3,5,7,17,9,19,8,18)(4,10,14,27,12,15,13,26)(6,21,22,32,24,33,23,16)(11,28,30,31,29,34,20,25);(3,6)(4,11,12,29)(5,16)(7,22)(8,23)(9,24)(10,25,15,31)(13,20,14,30)(17,21)(18,33)(19,32)(26,34,27,28)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265547);