Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#18334 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3126 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);(4,10)(5,15)(6,19)(7,16)(11,25)(12,29)(13,22)(14,27)(17,21)(18,20)(23,32)(26,28)(30,33)(31,34);(3,4)(5,11)(6,12)(7,22)(8,14)(9,13)(10,16)(15,26)(17,30)(18,29)(19,28)(20,33)(21,25)(23,34)(24,31)(27,32);(3,5,9,18)(4,11,13,29)(6,16,21,7)(8,15,24,20)(10,25,22,12)(14,26,31,33)(17,32,19,23)(27,28,34,30);(3,6,8,17)(4,12,14,30)(5,16,15,32)(7,20,23,18)(9,21,24,19)(10,26,27,11)(13,25,31,28)(22,33,34,29)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254078);