Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9380 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,13,6,9)(7,12)(8,11)(10,15,17,14)(16,18)(19,20,23,27,24,28,22,26)(21,29,31,25,32,33,30,34);(3,5,7,6)(4,9)(8,15,16,14)(10,11)(12,13)(17,18)(19,21)(20,25)(22,31)(23,30)(24,32)(26,33)(27,29)(28,34);(3,6,7,5)(4,10,12,17)(8,14,16,15)(9,11,13,18)(19,21)(20,25)(22,30)(23,31)(24,32)(26,29)(27,33)(28,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107513);