Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19491 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3106 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1764 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);(3,4,8,11)(5,12,14,17)(6,10,7,16)(9,15,18,13)(19,20,24,28)(21,25,30,34)(22,27,23,32)(26,31,33,29);(3,5,6,15,8,14,7,13)(4,9,16,17,11,18,10,12)(19,21)(20,25)(22,31)(23,29)(24,30)(26,32)(27,33)(28,34);(19,21)(20,26)(22,29)(23,31)(24,30)(25,32)(27,34)(28,33);(3,6,8,7)(4,10,11,16)(5,13,14,15)(9,17,18,12)(19,22,24,23)(20,27,28,32)(21,29,30,31)(25,26,34,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261757);