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×#10804 |
| 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)(9,17,20,30)(10,21,23,14)(11,15)(12,16)(13,19,26,34)(18,31)(22,36,37,27)(24,28)(25,32)(29,33)(35,38);(7,9)(8,13,16,26)(10,18)(11,19)(12,20)(14,25,21,35)(15,17,28,30)(22,32)(23,33)(24,34)(27,29,36,31)(37,38);(4,6)(7,10,11,22,12,23,24,37)(8,14,15,27,16,21,28,36)(9,18,19,32,20,33,34,38)(13,25,17,29,26,35,30,31)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260692);