Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 133 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(18,21)(19,23)(22,24);(2,5)(6,7)(9,10,12,14)(11,15,16,13)(18,22)(21,24);(9,11,12,16)(10,13,14,15)(17,18,20,22)(19,24,23,21);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|b^2=gh,c^2=g,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #258963, #258964, #258965, … |
| maximal normal subgroup of | #258963, #258964, #258965, … |
| maximal quotient of | #326553, #326562, #326571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22244);