Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#2436 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C22×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2815 conjugacy classes of subgroups; only the 647 normal subgroups are shown.
This group has 2815 conjugacy classes of subgroups — too many to draw. Only its 647 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);(8,10)(11,12,15,18)(13,17,14,16);(4,6)(8,10)(11,13,15,14)(12,16,18,17);(3,4)(5,6)(7,8,9,10)(11,12)(13,16)(14,17)(15,18);(11,14,15,13)(12,17,18,16)76 conjugacy classes — view the full character table.
| maximal subgroup of | #98500, #111893, #112545, … |
| maximal normal subgroup of | #98500, #111893, #112545, … |
| maximal quotient of | C2×#13881, C2×#13922, C2×#13978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27689);