Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) = C29×C4 (order 2048) · Inn(G) = C23 · Out(G) = C26×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C23 |
| commutator, socle, prime radical | C23 | C22×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 83 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:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,22,19,15,23,25,17,21,26,28,16,24,20,27);(6,9)(10,12)(13,15,18,25,19,21,23,28,17,24,26,27,16,14,20,22);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,19)(14,15,21,24)(18,20,26,23)(22,25,28,27)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=d,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13105, #494765, #494766, … |
| maximal normal subgroup of | C2×#13105, #494765, #494766, … |
| maximal quotient of | #140067, #140074, #140078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6093);