Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29021 |
| 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 | |
1724 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);(3,4,6,8)(5,9,10,7)(11,12,14,15)(13,17,18,16)(19,20,23,26)(21,28,31,24)(22,27,29,25)(30,33,32,34);(3,5,6,10)(4,7,8,9)(12,15)(16,17)(19,21,29,30,23,31,22,32)(20,24,25,34,26,28,27,33);(12,15)(16,17)(19,21,23,31)(20,24,26,28)(22,30,29,32)(25,33,27,34);(4,8)(7,9)(11,13,14,18)(12,16,15,17)(19,22,23,29)(20,25,26,27)(21,30,31,32)(24,33,28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367700);