Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#17563 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,8)(5,6)(9,10)(11,12)(13,21,24,17)(14,18)(15,19)(16,20)(22,26,23,25);(3,4)(5,8)(6,10)(7,9)(11,13)(12,17)(14,22)(15,23)(16,24)(18,25)(19,26)(20,21);(3,5,7,6)(4,8,9,10)(11,14,16,15)(12,18,20,19)(13,22,24,23)(17,25,21,26);(11,15,16,14)(12,19,20,18)(13,23,24,22)(17,26,21,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250529);