Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
539 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,8,10,7,11,6,15)(5,12,16,17,13,18,14,9);(3,5,6,14,7,13,8,16)(4,9,10,12,11,17,15,18);(5,13)(6,8)(10,15)(12,18)46 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2069⋊C3 |
| maximal subgroup of | C4×#2069, #377688, #377934, … |
| maximal normal subgroup of | C4×#2069, #377688, #377934, … |
| maximal quotient of | C2×#10603, C2×#10607, C2×#10605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16608);