Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#30361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1902 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);(4,7)(8,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,26)(22,27)(23,28)(24,29)(25,31)(30,34)(32,33);(3,4,6,7)(5,8,9,10)(19,21,22,30,23,31,24,32)(20,25,27,33,28,26,29,34);(3,5)(4,8)(6,9)(7,10)(12,15)(16,18)(19,21,23,31)(20,26,28,25)(22,32,24,30)(27,33,29,34);(5,9)(8,10)(11,13,14,17)(12,16,15,18)(19,22,23,24)(20,27,28,29)(21,30,31,32)(25,33,26,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341107);