Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#1662 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#562 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
479 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 479 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,8,12)(5,13,15,18)(6,11,7,10)(9,16,17,14)(19,20,23,26)(21,27,29,34)(22,30,32,25)(24,31,33,28);(3,5)(4,9)(6,16)(7,14)(8,15)(10,18)(11,13)(12,17)(19,21,23,29)(20,24,26,33)(22,31,32,28)(25,34,30,27);(3,5)(4,9)(6,14)(7,16)(8,15)(10,13)(11,18)(12,17)(22,32)(25,30)(27,34)(28,31)40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7525, C4×#1662, #209154, … |
| maximal normal subgroup of | C2×#7525, C4×#1662, #209154, … |
| maximal quotient of | C2×#7787, C2×#7762, C2×#7795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11973);