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×#18333 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3056 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);(4,9)(5,13)(6,17)(10,23)(11,21)(12,18)(14,29)(15,16)(19,32)(22,27)(24,30)(25,34)(26,33)(28,31);(3,4,7,12)(5,10,15,25)(6,18,17,9)(8,11,20,26)(13,27,16,24)(14,30,31,22)(19,33,32,21)(23,28,34,29);(6,19)(9,21)(14,29)(17,32)(18,33)(22,34)(23,30)(28,31);(3,5,8,16)(4,10,11,24)(6,14,19,29)(7,15,20,13)(9,22,21,34)(12,25,26,27)(17,31,32,28)(18,30,33,23)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254077);