Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27505 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,8)(5,9,6,10)(11,12,15,18)(13,17,14,16)(19,20,23,26)(21,28,31,25)(22,32,24,27)(29,33,30,34);(4,8)(9,10)(11,13,15,14)(12,16,18,17)(19,21,23,31)(20,25,26,28)(22,30,24,29)(27,34,32,33);(3,5,7,6)(4,9,8,10)(11,14,15,13)(12,17,18,16)(20,26)(21,29)(22,24)(25,33)(28,34)(30,31);(3,6,7,5)(4,9,8,10)(12,18)(16,17)(19,22,23,24)(20,27,26,32)(21,30,31,29)(25,34,28,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358404);