Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7262 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #340 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C23×C8 | C22 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
187 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,17,14,19,13,18)(11,20,22,16,23,15,21,24);(2,5,7,6)(10,15)(11,21)(12,13)(16,17)(18,24)(19,20)(22,23)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=h,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,c^b=cg,d^b=dh,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7264, #127526, #127557, … |
| maximal normal subgroup of | C2×#7264, #127526, #127557, … |
| maximal quotient of | #93051, #93054, #93055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 252);