Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#17573 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3372 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)(5,11,13,8)(6,14)(7,10)(9,15)(12,16,18,17)(19,20)(21,29,31,34)(22,27)(23,28)(24,26)(25,30,33,32);(3,5)(4,8)(6,12)(7,13)(9,16)(10,11)(14,17)(15,18)(19,21)(20,25)(22,30)(23,32)(24,31)(26,33)(27,29)(28,34);(3,6,7,15)(4,9,10,14)(5,12,13,18)(8,16,11,17)(20,26)(25,33)(27,28)(29,34);(19,22,24,23)(20,27,26,28)(21,30,31,32)(25,29,33,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250540);