Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#2171 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C8×D4 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 199 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 — 4 generators on 34 points:
(1,2);(3,4,8,14,9,13,26,32,10,15,27,17,7,23,25,33)(5,16,19,21,18,11,28,22,12,29,31,20,24,30,34,6);(3,5)(4,11)(6,17)(7,24)(8,19)(9,18)(10,12)(13,29)(14,22)(15,30)(16,23)(20,32)(21,33)(25,34)(26,28)(27,31);(3,6,8,16,9,21,26,11,10,22,27,29,7,20,25,30)(4,12,14,31,13,24,32,34,15,5,17,19,23,18,33,28)88 conjugacy classes — view the full character table.
| maximal subgroup of | #128694, #128695, #134990, … |
| maximal normal subgroup of | #128694, #128695, #134990, … |
| maximal quotient of | C2×#12038, #483490, C2×#12050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26428);