Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27072 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2184 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);(3,4,7,8)(5,9,6,10)(11,12,14,15)(13,17,18,16)(19,20)(21,27)(22,25)(23,26)(24,29)(28,32)(30,34)(31,33);(4,8)(9,10)(12,15)(16,17)(19,21)(20,24)(22,29)(23,28)(25,27)(26,34)(30,31)(32,33);(3,5,7,6)(4,9,8,10)(11,13,14,18)(12,16,15,17)(20,25)(21,28)(23,31)(24,32)(27,34)(29,30);(3,6,7,5)(4,9,8,10)(12,15)(16,17)(20,25)(24,27)(26,33)(32,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348593);