Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#3025 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, Frattini, cosocle | C23 | C25 |
| socle | C22 | C2×#120 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
2543 conjugacy classes of subgroups; only the 559 normal subgroups are shown.
This group has 2543 conjugacy classes of subgroups — too many to draw. Only its 559 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 — 5 generators on 18 points:
(1,2);(8,14)(15,18);(3,4)(5,8)(6,9)(7,10)(11,15)(12,14)(13,16)(17,18);(3,5)(4,8)(6,11)(7,12)(9,15)(10,14)(13,17)(16,18);(3,6,7,13)(4,9,10,16)(5,11,12,17)(8,15,14,18)64 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53382);