Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#29709 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3459 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4246 conjugacy classes of subgroups; only the 626 normal subgroups are shown.
This group has 4246 conjugacy classes of subgroups — too many to draw. Only its 626 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,5)(8,9)(11,12)(13,15)(14,16)(17,18);(3,4)(5,7)(6,8)(9,10)(11,13)(12,15)(14,17)(16,18);(3,5,7,4)(6,8,10,9)(15,18);(3,6,7,10)(4,8,5,9)74 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6491478, C2×#6542240 |
| maximal normal subgroup of | C2×#6491478, C2×#6542240 |
| maximal quotient of | #72318118 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342589);