Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#1410 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #353 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4⋊Q8 | C23 |
| prime radical | C2×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
225 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 225 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) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4,5,10,7,11,14,18)(6,8,12,16,15,9,13,17);(4,8)(5,12)(6,15)(9,11)(13,14)(16,17)34 conjugacy classes — view the full character table.
| maximal subgroup of | #94781, #94798, #94804, … |
| maximal normal subgroup of | #94781, #94798, #94804, … |
| maximal quotient of | C2×#7354, C2×#7371, #149690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6575);