Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29009 |
| 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);(3,4,6,8)(5,9,10,7)(11,12,14,15)(13,17,18,16)(19,20,23,26)(21,28,22,25)(24,27,29,32)(30,34,31,33);(3,5,6,10)(4,7,8,9)(12,15)(16,17)(19,21,29,31,23,22,24,30)(20,25,32,34,26,28,27,33);(12,15)(16,17)(19,22,23,21)(20,25,26,28)(24,30,29,31)(27,34,32,33);(4,8)(7,9)(11,13,14,18)(12,16,15,17)(20,26)(25,28)(27,32)(33,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367673);