Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#17840 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C22×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,8)(6,9)(10,15)(11,16)(12,18)(14,17)(19,20,23,26,24,27,22,32)(21,28,31,33,29,34,30,25);(3,4,9,16,7,11,6,8)(5,12,17,15,13,10,14,18)(19,21,23,31,24,29,22,30)(20,25,32,34,27,33,26,28);(3,5)(4,10)(6,14)(7,13)(8,15)(9,17)(11,12)(16,18)(21,29)(25,33)(28,34)(30,31);(5,13)(10,12)(14,17)(15,18)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251579);