Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#23360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,13,16,9)(6,17,20,31)(7,11)(8,12)(10,19,27,33)(14,25)(15,28,21,24)(18,23)(22,26)(29,32)(30,34);(3,5)(4,9)(6,18)(7,21)(8,16)(10,25)(11,28)(12,13)(14,19)(15,22)(17,29)(20,32)(23,31)(24,26)(27,34)(30,33);(3,6,7,19,8,20,22,33)(4,10,26,31,12,27,11,17)(5,14,15,18,16,30,21,32)(9,23,28,25,13,29,24,34);(4,11,12,26)(9,24,13,28)(10,17,27,31)(23,34,29,25)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314827);