Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#9634 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #389 |
| prime radical | C24 | #125 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,8,11,7,10,6,15)(5,12,13,9,14,18,16,17)(20,25)(21,28)(22,23)(26,33)(27,31)(29,30);(3,5,8,13,7,14,6,16)(4,9,11,18,10,17,15,12)(19,20)(21,26)(22,31)(23,25)(24,27)(28,32)(29,34)(30,33);(4,10)(9,17)(11,15)(12,18)(19,21,22,29,24,30,23,28)(20,26,31,34,27,33,25,32)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111189);