Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,23,21,20);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,21)(18,23,24,20);(1,3,5,4)(2,7,6,8)(17,20,21,18,22,23,19,24);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(18,24)(20,23)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,c^b=ch,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #293631, #293637, #293643, … |
| maximal normal subgroup of | #293631, #293637, #293643, … |
| maximal quotient of | #392533, #393180, #393278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26275);