Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20105 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×D42 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,8)(6,14)(9,15)(10,11)(12,18)(16,17)(19,20)(21,25)(22,29)(23,32)(24,27)(26,28)(30,34)(31,33);(3,4,7,11)(5,9,13,17)(6,10,14,8)(12,16,18,15)(19,21,24,23)(20,25,27,32)(22,30,31,28)(26,29,34,33);(3,5,7,13)(4,9,11,17)(6,12,14,18)(8,15,10,16)(19,22,24,31)(20,26,27,34)(21,28,23,30)(25,33,32,29);(4,10)(6,14)(8,11)(9,16)(12,18)(15,17)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266631);