Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9938 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#9941 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1668 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)(5,12)(6,10)(7,15)(8,11)(9,13)(14,18)(16,17)(19,20)(21,24)(22,25)(23,26)(27,30)(28,31)(29,33)(32,34);(3,5)(4,9)(6,13)(7,16)(8,14)(10,18)(11,17)(12,15)(19,21,23,28)(20,24,26,31)(22,27,29,34)(25,32,33,30);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,12,17,18)(19,22)(20,25)(21,27)(23,29)(24,30)(26,33)(28,34)(31,32)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142122);