Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#30843 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,7,10)(5,6,9,8)(11,12,14,19)(13,21,23,26)(15,17,16,20)(18,24,22,25);(3,5,7,9)(4,8,10,6)(11,12)(13,22)(14,19)(15,20)(16,17)(18,23)(21,24)(25,26);(3,6,7,8)(4,9,10,5)(12,17)(15,16)(18,26)(19,20)(21,22)(24,25);(4,10)(5,9)(11,13)(12,18)(14,23)(15,25)(16,24)(17,26)(19,22)(20,21)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409244);