Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#2235 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
315 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,15,19)(13,16,22,25)(14,23,21,18)(17,26,24,20);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(12,16)(13,20)(14,21)(17,18)(19,25)(22,26)(23,24)58 conjugacy classes — view the full character table.
| maximal subgroup of | #137429, #137434, #138972, … |
| maximal normal subgroup of | #137429, #137434, #138972, … |
| maximal quotient of | C2×#12866, C2×#12920, C2×#12993, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26628);