Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×C4×Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle, prime radical | C23 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4,5,6);(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(23,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24);(7,23,15,31)(8,24,16,32)(9,25,17,33)(10,26,18,34)(11,27,19,35)(12,28,20,36)(13,29,21,37)(14,30,22,38)88 conjugacy classes — view the full character table.
| maximal subgroup of | #136688, #136692, #136722, … |
| maximal normal subgroup of | #136688, #136692, #136722, … |
| maximal quotient of | C2×C8×Q32, C2×#12072, C2×#12083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26434);