Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#23321 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(12,15)(13,17)(19,20,23,26,22,24,30,31)(21,27,29,34,28,25,32,33);(3,5)(4,7)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18)(20,24)(21,28)(26,31)(29,32);(4,8)(5,9)(11,13,14,17)(12,16,15,18)(19,21,22,28)(20,25,24,27)(23,29,30,32)(26,33,31,34);(4,8)(7,10)(12,15)(16,18)(21,28)(25,27)(29,32)(33,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314672);