Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32696 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1517 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);(3,4,7,12,8,11,22,28)(5,13,16,29,15,9,21,24)(6,17,20,33,19,10,25,27)(14,30,18,32,31,23,34,26);(3,5)(4,9)(6,18)(7,21)(8,15)(10,26)(11,13)(12,29)(14,20)(16,22)(17,32)(19,34)(23,27)(24,28)(25,31)(30,33);(3,6)(4,10,11,17)(5,14,15,31)(7,20)(8,19)(9,23)(12,27,28,33)(13,30)(16,18,21,34)(22,25)(24,26)(29,32);(4,11)(7,22)(10,17)(16,21)(18,34)(20,25)(24,29)(26,32)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421132);