Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#10817 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #382 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1818 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)(9,19,22,14)(10,24,26,32)(11,17)(12,18)(13,16)(15,25,31,27)(20,34)(21,30,23,29)(28,35)(33,36)(37,38);(7,9,12,23,13,22,11,21)(8,14,18,30,16,19,17,29)(10,20,27,36,26,37,25,35)(15,28,32,34,31,33,24,38);(4,6)(7,10)(8,15,16,31)(9,20)(11,25)(12,27)(13,26)(14,28,19,33)(17,24,18,32)(21,35)(22,37)(23,36)(29,38,30,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260721);