Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#538 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3484 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);(3,4,6,12,8,11,17,25)(5,10,15,23,13,24,27,34)(7,9,16,21,18,20,30,32)(14,19,28,31,26,22,29,33);(4,9)(5,13)(6,16)(10,22)(11,20)(14,26)(15,29)(17,30)(19,24)(23,34)(27,28)(31,33);(3,5)(4,10)(6,15)(7,14)(8,13)(9,19)(11,24)(12,23)(16,28)(17,27)(18,26)(20,22)(21,31)(25,34)(29,30)(32,33);(4,11)(5,14)(9,20)(10,22)(12,25)(13,26)(15,28)(19,24)(21,32)(23,33)(27,29)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249843);