Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12727 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12729 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#519 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,12,8,13,20,28)(5,14,17,29,16,10,24,26)(6,9,19,23,18,11,21,27)(15,25,31,34,30,22,32,33);(4,9)(6,18)(7,20)(10,22)(11,13)(12,23)(14,25)(15,30)(17,24)(26,33)(27,28)(29,34);(3,5)(4,10,13,14)(6,15)(7,17)(8,16)(9,22,11,25)(12,26,28,29)(18,30)(19,31)(20,24)(21,32)(23,33,27,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400663);