Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#2229 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
271 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12,14,17)(13,21,22,39)(15,27,25,40)(16,30,26,20)(18,34,32,23)(19,36,33,29)(24,41,31,37)(28,38,42,35);(3,5,6,10)(4,7,8,9)(12,17)(18,32)(19,33)(20,30)(21,39)(27,40)(35,38)(37,41);(11,13,14,22)(12,18,17,32)(15,28,25,42)(16,31,26,24)(19,37,33,41)(20,38,30,35)(21,40,39,27)(23,36,34,29)58 conjugacy classes — view the full character table.
| maximal subgroup of | #138833, #138839, C4×#2229, … |
| maximal normal subgroup of | #138833, #138839, C4×#2229, … |
| maximal quotient of | C2×#12814, C2×#12936, C2×#12938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26613);