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×#17576 |
| 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)(5,12,15,9)(6,16)(7,10)(8,11)(13,17,14,18)(19,20)(21,29,31,34)(22,27)(23,28)(24,26)(25,30,33,32);(3,5)(4,9)(6,13)(7,14)(8,15)(10,17)(11,12)(16,18)(19,21)(20,25)(22,30)(23,32)(24,31)(26,33)(27,29)(28,34);(3,6,8,7)(4,10,11,16)(5,13,15,14)(9,17,12,18)(20,26)(25,33)(27,28)(29,34);(3,7,8,6)(4,10,11,16)(5,14,15,13)(9,17,12,18)(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, 6250543);