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) = C28×C4 |
| direct product | C2×#2501 |
| 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 | |
2111 conjugacy classes of subgroups; only the 647 normal subgroups are shown.
This group has 2111 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);(4,5)(8,9)(11,12,15,17)(13,14,16,18);(3,4,6,5)(7,8,10,9)(11,13)(12,14)(15,16)(17,18);(7,8,10,9)(11,14)(12,16)(13,17)(15,18);(3,5,6,4)(7,9,10,8)76 conjugacy classes — view the full character table.
| maximal subgroup of | #98546, #111894, #112544, … |
| maximal normal subgroup of | #98546, #111894, #112544, … |
| maximal quotient of | C2×#13880, C2×#13904, C2×#13996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27744);