Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11086 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #397 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2702 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);(7,8,12,18,13,16,11,17)(9,14,22,30,20,31,21,29)(10,23,27,33,25,15,26,34)(19,28,36,37,35,38,24,32);(3,4)(5,6)(7,9)(8,14)(10,24,25,36)(11,21)(12,22)(13,20)(15,32,23,37)(16,31)(17,29)(18,30)(19,26,35,27)(28,33,38,34);(4,6)(7,10)(8,15)(9,19)(11,27)(12,26)(13,25)(14,28)(16,23)(17,34)(18,33)(20,35)(21,36)(22,24)(29,37)(30,32)(31,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264953);