Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17615 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3505 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3364 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);(4,9,11,23)(6,18)(10,22,25,33)(12,24,13,19)(15,29)(20,32)(26,34,27,30)(28,31);(3,4)(5,10)(6,19)(7,12)(8,13)(9,20)(11,21)(14,25)(15,30)(16,26)(17,27)(18,23)(22,31)(24,32)(28,34)(29,33);(3,5,7,16)(4,10,12,26)(6,15,18,29)(8,17,21,14)(9,22,24,34)(11,25,13,27)(19,30,23,33)(20,31,32,28);(5,14)(10,25)(15,28)(16,17)(22,33)(26,27)(29,31)(30,34)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250646);