Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#1298 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | #95 |
| prime radical | C25 | Q8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
469 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 469 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) = 64 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,14,19,16,20,15,18)(13,21,22,17,24,25,23,26);(3,5)(4,7,8,9)(6,10)(11,13,15,23,16,24,14,22)(12,17,19,25,20,26,18,21)76 conjugacy classes — view the full character table.
| maximal subgroup of | #94300, #94302, C4×#1298, … |
| maximal normal subgroup of | #94300, #94302, C4×#1298, … |
| maximal quotient of | C2×#7044, #95296, #95298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1041);