Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
658 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 658 conjugacy classes of subgroups — too many to draw. Only its 121 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,22)(13,16)(14,17)(15,19)(20,23)(21,24);(2,5)(6,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(1,3)(2,6)(4,7)(5,8)(9,11,14,21)(10,15,17,23)(12,20,13,19)(16,24,22,18);(3,7)(6,8)Polycyclic presentation:
2^8|c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,g^b=gh,d^c=de,f^c=fh,g^c=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | #248771, #248772, #248776, … |
| maximal normal subgroup of | #248771, #248772, #248776, … |
| maximal quotient of | #326686, #326688, #326694, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22606);