Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24000 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3916 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)(5,9)(6,16)(7,20)(8,12)(10,19)(11,22)(13,28)(14,30)(15,25)(17,33)(18,27)(21,26)(23,29)(24,32)(31,34);(3,5,8,15)(4,9,12,25)(6,17)(7,14,22,32)(10,23)(11,24,20,30)(13,18)(16,28)(19,31)(21,29)(26,34)(27,33);(3,6)(4,10)(5,13)(7,21)(8,18)(9,23)(11,27)(12,26)(14,31)(15,17)(16,20)(19,22)(24,33)(25,34)(28,30)(29,32);(7,22)(11,20)(14,32)(16,27)(19,21)(24,30)(28,33)(29,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358926);