Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17700 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3332 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);(3,4,7,12,8,11,6,10)(5,13,14,18,15,9,16,17)(19,20)(21,28)(22,26)(23,31)(24,27)(25,29)(30,33)(32,34);(3,5)(4,9)(6,14)(7,16)(8,15)(10,17)(11,13)(12,18)(19,21)(20,25)(22,29)(23,32)(24,30)(26,33)(27,34)(28,31);(19,22,24,23)(20,26,27,31)(21,29,30,32)(25,33,34,28);(3,6,8,7)(4,10,11,12)(5,14,15,16)(9,17,13,18)(19,22,24,23)(20,26,27,31)(21,29,30,32)(25,33,34,28)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251162);