Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#22478 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3646 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);(4,7)(8,10)(11,12)(13,17)(14,22,24,18)(15,19)(16,20)(21,26,23,25);(3,4)(5,8)(6,7)(9,10)(11,13,16,15)(12,17,20,19)(14,21,24,23)(18,25,22,26);(3,5,6,9)(4,8,7,10)(11,14)(12,18)(13,21)(15,23)(16,24)(17,25)(19,26)(20,22);(4,7)(8,10)(11,15,16,13)(12,19,20,17)(14,23,24,21)(18,26,22,25)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286440);