Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#7781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#2733 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(7,10)(11,12,15,18)(13,16,14,17)(20,23)(21,25)(24,26);(3,4,6,8)(5,7,9,10)(12,16)(13,14)(17,18)(21,25)(23,26);(3,5,6,9)(4,7,8,10)(12,17)(13,14)(16,18)(19,20,22,24)(21,26,25,23)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 72946);