Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17715 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,6,10,7,9,15,17)(5,11,12,18,13,8,14,16)(19,20)(21,28)(22,27)(23,31)(24,26)(25,29)(30,34)(32,33);(3,5,7,13)(4,8,9,11)(6,14,15,12)(10,18,17,16)(19,21,24,30)(20,25,26,33)(22,29,23,32)(27,34,31,28);(4,9)(8,11)(10,17)(16,18)(20,26)(25,33)(27,31)(28,34);(19,22,24,23)(20,27,26,31)(21,29,30,32)(25,34,33,28)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251177);