Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 123 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)(11,15)(12,14)(13,19)(16,18)(17,21)(20,23)(22,24);(2,5)(6,8)(9,11,15,10,16,12,14,18)(13,19,22,17,23,20,21,24);(1,3,4,7)(2,6,5,8)(9,12,14,10,16,11,15,18)(13,20,21,17,23,19,22,24);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)Polycyclic presentation:
2^8|b^2=f,c^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=ch,g^d=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #273723, #273724, #273725, … |
| maximal normal subgroup of | #273723, #273724, #273725, … |
| maximal quotient of | #350990, #350991, #350996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26051);