Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#19629 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,7)(5,9)(8,10)(12,15)(13,16)(17,18);(3,4,5,7,6,8,9,10);(11,12,16,18,14,17,13,15);(4,8)(7,10)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6357200, #72224780, #72224784, … |
| maximal normal subgroup of | C2×#6357200, #72224780, #72224784, … |
| maximal quotient of | C2×#233367, #72342610 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264306);