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×#1411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #353 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4⋊Q8 | C23 |
| prime radical | C23 | #99 |
| 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 | |
199 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 199 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);(3,4,6,11,8,13,20,30)(5,14,15,28,17,25,32,33)(7,12,22,29,23,10,19,27)(9,18,24,21,26,31,34,16);(3,5,8,17)(4,9,13,26)(6,18,20,31)(7,21,23,16)(10,25,12,14)(11,28,30,33)(15,19,32,22)(24,29,34,27)34 conjugacy classes — view the full character table.
| maximal subgroup of | #94777, #94796, #94802, … |
| maximal normal subgroup of | #94777, #94796, #94802, … |
| maximal quotient of | C2×#7355, C2×#7370, #149692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6576);