Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11158 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #397 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8)(9,18)(10,23,27,38)(11,28)(12,16)(13,17)(14,20)(15,26,31,25)(19,34,24,29)(21,30)(22,32)(33,36,37,35);(7,9,13,22)(8,14,17,30)(10,24,27,19)(11,21,12,20)(15,29,31,34)(16,32,28,18)(23,37,38,33)(25,35,26,36);(4,6)(7,10,11,25,13,27,12,26)(8,15,16,23,17,31,28,38)(9,19,20,35,22,24,21,36)(14,29,18,33,30,34,32,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265112);