Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#17398 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ 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);(3,4,6,12,8,11,18,25)(5,10,15,24,13,23,26,34)(7,9,17,21,14,20,28,32)(16,19,30,31,27,22,29,33);(4,9)(5,13)(6,17)(10,22)(11,20)(15,29)(16,27)(18,28)(19,23)(24,34)(26,30)(31,33);(3,5,14,27)(4,10,20,22)(6,15,28,29)(7,16,8,13)(9,19,11,23)(12,24,32,33)(17,30,18,26)(21,31,25,34);(4,11)(9,20)(10,23)(12,25)(19,22)(21,32)(24,34)(31,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249827);