Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17597 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3519 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3450 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);(4,9,11,22)(5,14)(6,17)(10,24,26,33)(12,23,13,18)(15,28)(16,31)(19,32)(21,25,34,27)(29,30);(3,4)(5,10)(6,18)(7,12)(8,13)(9,19)(11,20)(14,27)(15,25)(16,24)(17,22)(21,29)(23,32)(26,28)(30,34)(31,33);(3,5)(4,10)(6,16)(7,15)(8,14)(9,21)(11,26)(12,25)(13,27)(17,30)(18,24)(19,29)(20,28)(22,34)(23,33)(31,32);(5,15)(10,25)(14,28)(16,30)(21,33)(24,34)(26,27)(29,31)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250578);