Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#1669 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
395 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 395 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) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,15,18)(13,19,22,26)(14,23,20,17)(16,24,25,21);(11,13,20,24,15,22,14,21)(12,16,23,26,18,25,17,19);(4,7)(5,10)(8,9)58 conjugacy classes — view the full character table.
| maximal subgroup of | #136880, #136881, #136887, … |
| maximal normal subgroup of | #136880, #136881, #136887, … |
| maximal quotient of | C2×#8644, C2×#9219, C2×#9561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12088);