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×#1663 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
415 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 415 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) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,9,13)(5,14,16,17)(6,15,10,18)(7,12,8,11)(19,20,24,22)(21,27,30,34)(23,31,33,26)(25,32,28,29);(3,5)(4,10)(6,13)(7,18)(8,15)(9,16)(11,17)(12,14)(19,21,24,30)(20,25,22,28)(23,32,33,29)(26,34,31,27);(3,6,9,10)(4,5,13,16)(7,17,8,14)(11,15,12,18)(19,22,24,20)(21,28,30,25)(23,26,33,31)(27,32,34,29)40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7526, C4×#1663, #209161, … |
| maximal normal subgroup of | C2×#7526, C4×#1663, #209161, … |
| maximal quotient of | C2×#7763, C2×#7792, C2×#7798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11977);