Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2032 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
717 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 717 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)(5,8)(6,13,15,9)(7,10)(11,18,14,16)(12,17);(3,5)(4,8)(6,14,15,11)(7,12)(9,16,13,18)(10,17);(3,6)(4,9)(5,11)(7,15)(8,16)(10,13)(12,14)(17,18)46 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2032⋊C3 |
| maximal subgroup of | C4×#2032, #375943, #391000, … |
| maximal normal subgroup of | C4×#2032, #375943, #391000, … |
| maximal quotient of | C2×#10390, C2×#10392, C2×#10391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16461);