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×#10019 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3109 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1496 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)(7,11)(8,12)(10,14)(15,17)(16,18)(20,24)(22,29)(25,33)(26,30)(27,31)(32,34);(4,9)(6,7)(10,13)(11,12)(14,16)(17,18)(19,20)(21,25)(22,30)(23,26)(24,29)(27,32)(28,34)(31,33);(3,5,8,15)(4,10,12,18)(6,14,7,16)(9,17,11,13)(19,21,23,28)(20,25,26,34)(22,31,29,27)(24,32,30,33)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142623);