Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#1407 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #353 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
208 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4)(5,13,15,28)(6,17,18,33)(7,20,22,11)(8,12)(9,14,23,30)(10,19)(16,26,21,29)(24,31,25,32)(27,34);(3,5,8,16)(4,9,12,25)(6,14,18,32)(7,21,22,15)(10,26,27,13)(11,23,20,24)(17,29,33,28)(19,30,34,31)28 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1407, #151919, #151955, … |
| maximal normal subgroup of | C4×#1407, #151919, #151955, … |
| maximal quotient of | C2×#7353, C2×#7372, #149686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6564);