Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18345 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,12)(5,13,15,17)(6,10,7,11)(9,16,18,14)(20,24)(22,29)(25,33)(26,30)(27,31)(32,34);(3,5)(4,9)(6,14)(7,16)(8,15)(10,17)(11,13)(12,18)(19,20)(21,25)(22,30)(23,26)(24,29)(27,32)(28,34)(31,33);(3,6,8,7)(4,10,12,11)(5,14,15,16)(9,17,18,13)(19,21,23,28)(20,25,26,34)(22,31,29,27)(24,32,30,33);(3,6,8,7)(4,11,12,10)(5,14,15,16)(9,13,18,17)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254090);