Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#32034 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2222 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)(11,12)(13,17)(14,16)(15,18)(19,20,24,23)(21,25,28,31)(22,26,30,32)(27,33,34,29);(3,5,7,4)(6,9,10,8)(11,13,14,18)(12,15,16,17)(19,21)(20,25)(22,29)(23,31)(24,28)(26,27)(30,33)(32,34);(3,6,7,10)(4,8,5,9)(12,16)(15,17)(19,22)(20,26)(21,27)(23,32)(24,30)(25,33)(28,34)(29,31);(4,5)(8,9)(13,18)(15,17)(19,20,24,23)(21,25,28,31)(22,26,30,32)(27,33,34,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7414681);