Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10512 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,11)(5,12,13,9)(6,8,15,10)(14,17,18,16)(19,20)(21,29)(22,27)(23,26)(24,28)(25,31)(30,34)(32,33);(4,8)(5,13)(6,15)(9,17)(10,11)(12,16)(19,21)(20,25)(22,31)(23,32)(24,30)(26,34)(27,29)(28,33);(3,5,6,14,7,13,15,18)(4,9,10,17,11,12,8,16)(20,26)(21,30)(23,24)(27,28)(31,32)(33,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146002);