Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C4×#3495 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4);(7,11)(9,12)(13,14,18,21)(15,16,23,19)(17,25,26,20)(22,27,28,24);(5,6,8,10)(7,9,11,12)(13,15,18,23)(14,19,21,16)(17,22,26,28)(20,27,25,24);(5,7)(6,9)(8,11)(10,12)(13,16)(14,15)(17,24)(18,19)(20,22)(21,23)(25,28)(26,27);(13,15)(14,16)(17,22)(18,23)(19,21)(20,24)(25,27)(26,28);(13,17,18,26)(14,20,21,25)(15,22,23,28)(16,24,19,27)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532870);