Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#30489 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,13,17,14,9,12,16)(10,11,21,19,15,18,20,22);(7,9,14,8)(10,19,15,22)(11,20,18,21)(12,16,13,17);(4,6)(7,10,14,15)(8,11,9,18)(12,20,13,21)(16,22,17,19);(4,6)(7,11)(8,15)(9,10)(12,19)(13,22)(14,18)(16,20)(17,21)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408095);