Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32007 |
| 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 | |
2102 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)(5,6)(7,9)(8,10)(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)(4,8)(6,9)(7,10)(11,13)(12,15)(14,18)(16,17)(19,21,24,28)(20,25,23,31)(22,29,30,33)(26,27,32,34);(3,6,7,8)(4,5,9,10)(12,16)(15,17)(19,22)(20,26)(21,27)(23,32)(24,30)(25,33)(28,34)(29,31);(4,9)(5,10)(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, 7414595);