Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#2241 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,18,13)(14,15,24,26)(16,23,19,20)(17,25,22,21);(4,7)(8,10)(11,13,16,24,17,25,15,23,18,12,19,14,22,21,26,20);(3,5,6,9)(4,8,7,10)(11,14,15,13,22,23,16,21,18,24,26,12,17,20,19,25)76 conjugacy classes — view the full character table.
| maximal subgroup of | #137506, #137508, #137567, … |
| maximal normal subgroup of | #137506, #137508, #137567, … |
| maximal quotient of | C2×#12888, C2×#13148, C2×#13203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26683);