Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | Aut(G) = #72318105 (order 2048) · Inn(G) = #354 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#1414 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #354 |
| 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 | |
170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 170 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)(6,17,19,33)(7,21,23,11)(8,12)(9,14,24,31)(10,20)(16,30,22,29)(18,25,32,26)(27,34);(3,5)(4,9,12,26)(6,18)(7,22)(8,16)(10,13,27,30)(11,24,21,25)(14,19)(15,23)(17,28,33,29)(20,32)(31,34)34 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1414, #151940, #151943, … |
| maximal normal subgroup of | C4×#1414, #151940, #151943, … |
| maximal quotient of | C2×#7359, C2×#7366, #149705, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6589);