Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#2004 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1095 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 1095 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8)(9,15)(10,18)(11,14)(12,16)(13,20)(17,22)(19,21);(7,9,11,17)(8,12,14,21)(10,19,20,16)(13,22,18,15);(4,6)(10,20)(12,21)(13,18)(16,19)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2004, #359065, #375817, … |
| maximal normal subgroup of | C4×#2004, #359065, #375817, … |
| maximal quotient of | C2×#10081, C2×#10085, C2×#10086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16384);