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×#26878 |
| 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 | |
1664 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);(3,4,7,9)(5,10,6,8)(11,12)(13,16)(14,15)(17,18)(19,20,23,26)(21,28,31,25)(22,32,24,27)(29,33,30,34);(3,5,7,6)(4,8,9,10)(19,21)(20,25)(22,30)(23,31)(24,29)(26,28)(27,34)(32,33);(3,6,7,5)(4,8,9,10)(12,15)(16,18)(20,26)(21,29)(22,24)(25,33)(28,34)(30,31);(4,9)(8,10)(11,13,14,17)(12,16,15,18)(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, 6353127);