Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10144 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10163 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(4,9)(5,13)(6,7)(11,12)(14,17)(15,16)(19,20,23,26)(21,27,28,34)(22,30,31,25)(24,32,33,29);(3,4,8,12)(5,14,16,17)(6,11,7,9)(10,13,18,15)(22,31)(25,30)(27,34)(29,32);(3,5,8,16)(4,10,12,18)(6,15,7,13)(9,14,11,17)(19,21)(20,24)(22,32)(23,28)(25,34)(26,33)(27,30)(29,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144098);