Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18192 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18200 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3490 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,12)(5,13,16,29)(6,17,19,31)(7,10,21,25)(9,22,23,15)(11,26,28,20)(14,27,30,34)(18,24,32,33);(3,5)(4,9)(6,18)(7,14)(8,16)(10,24)(11,27)(12,23)(13,17)(15,20)(19,32)(21,30)(22,26)(25,33)(28,34)(29,31);(6,19)(11,28)(13,29)(15,22)(17,31)(18,32)(20,26)(27,34);(4,10)(5,14)(11,17)(12,25)(15,18)(16,30)(22,32)(28,31)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253665);