Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10023 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1426 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);(3,4,8,12)(5,13,15,17)(6,9,7,11)(10,16,18,14)(19,20,24,22)(21,27,30,34)(23,31,33,26)(25,32,28,29);(4,9)(6,7)(10,13)(11,12)(14,16)(17,18)(19,21)(20,25)(22,28)(23,32)(24,30)(26,34)(27,31)(29,33);(3,5,8,15)(4,10,12,18)(6,14,7,16)(9,17,11,13)(19,22,24,20)(21,28,30,25)(23,26,33,31)(27,32,34,29)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142624);