Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10139 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1214 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);(4,9)(6,7)(10,17)(11,12)(13,18)(14,16)(19,20,24,21)(22,29,30,34)(23,28,33,26)(25,32,27,31);(3,4)(5,13)(6,12)(7,9)(8,11)(10,14)(15,17)(16,18)(19,21,24,20)(22,27,30,25)(23,26,33,28)(29,32,34,31);(3,5,8,15)(4,10,11,18)(6,14,7,16)(9,13,12,17)(19,22)(20,25)(21,27)(23,32)(24,30)(26,34)(28,29)(31,33)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 143920);