Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18162 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18174 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3110 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);(3,4)(5,12)(6,10)(7,16)(8,11)(9,14)(13,17)(15,18)(19,20,23,26)(21,24,28,31)(22,25,29,32)(27,33,34,30);(3,5,8,15)(4,9,11,17)(6,14,7,13)(10,18,16,12)(19,21)(20,24)(22,27)(23,28)(25,30)(26,31)(29,34)(32,33);(19,22)(20,25)(21,27)(23,29)(24,30)(26,32)(28,34)(31,33);(3,6,8,7)(4,10,11,16)(5,13,15,14)(9,12,17,18)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253625);