Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C23 |
| commutator | C22 | C22×C16 |
| socle, prime radical | C23 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,16,20,17,21,24,27,18,15,19,22,25,23,26,28);(1,2,3,4)(6,9)(10,11)(13,15,16,22,17,23,24,28,18,14,19,20,25,21,26,27);(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,f^2=g,g^2=h,b^a=bd,c^a=ce112 conjugacy classes — view the full character table.
| maximal subgroup of | #92998, #93002, #93014, … |
| maximal normal subgroup of | #92998, #93002, #93014, … |
| maximal quotient of | #141842, #141849, #141861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6251);