Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2030 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
741 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 741 conjugacy classes of subgroups — too many to draw. Only its 175 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);(3,4,6,8)(5,9,10,7)(11,12)(13,16)(14,17)(15,18)(19,24)(20,23)(21,25)(22,26);(3,5,6,10)(4,7,8,9)(11,13)(12,16)(14,19)(15,20)(17,24)(18,23)(21,26)(22,25);(11,14,15,21)(12,17,18,25)(13,19,20,26)(16,22,23,24)52 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#2030⋊C3 |
| maximal subgroup of | #137090, #137092, #137121, … |
| maximal normal subgroup of | #137090, #137092, #137121, … |
| maximal quotient of | C2×#10342, C2×#10346, C2×#10343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16454);