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) = C25 |
| direct product | C2×#2061 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
565 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 565 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) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,7,9)(5,10,6,8)(11,12,14,18)(13,20,21,16)(15,17,23,19)(22,26,24,25);(3,5,7,6)(4,8,9,10)(11,13,14,21)(12,16,18,20)(15,24,23,22)(17,26,19,25);(3,6,7,5)(4,8,9,10)(12,17)(15,23)(16,25)(18,19)(20,26)(22,24)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137106, #137108, #137115, … |
| maximal normal subgroup of | #137106, #137108, #137115, … |
| maximal quotient of | C2×#10571, C2×#10568, C2×#10569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16547);