Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#29651 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3172 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);(3,4,7,10)(5,11,13,8)(6,9,14,16)(12,17,18,15)(19,20)(21,24)(22,25)(23,27)(26,31)(28,32)(29,34)(30,33);(3,5,7,13)(4,8,10,11)(6,12,14,18)(9,15,16,17)(19,21,23,29)(20,24,27,34)(22,30,31,28)(25,33,26,32);(3,6)(4,9)(5,12)(7,14)(8,15)(10,16)(11,17)(13,18)(20,25)(22,31)(24,32)(26,27)(28,30)(33,34);(6,14)(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, 6341896);