Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11143 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| 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 | |
2618 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)(7,8,12,16,13,18,11,17)(9,19,23,31,22,30,21,14)(10,24,27,15,28,32,26,33)(20,35,37,34,25,29,36,38);(7,9,11,21,13,22,12,23)(8,14,16,19,18,31,17,30)(10,25,26,37,28,20,27,36)(15,29,32,38,33,35,24,34);(4,6)(7,10,13,28)(8,15,18,33)(9,20,22,25)(11,27,12,26)(14,29,31,35)(16,24,17,32)(19,34,30,38)(21,37,23,36)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265034);