Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) = C24×#442 (order 1024) · Inn(G) = #442 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C2×D8 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
115 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 115 conjugacy classes of subgroups — too many to draw. Only its 53 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 — 3 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10,12,18,14,19,13,16)(11,20,21,24,22,15,23,17);(10,15)(11,21)(12,13)(16,17)(18,24)(19,20)(22,23);(1,3,4,8,5,2,6,7)(10,16,19,18)(11,22)(15,17,20,24)(21,23)Polycyclic presentation:
2^8|a^2=e,c^2=ef,d^2=f,e^2=g,f^2=h,b^a=bd,c^a=cf,d^a=df,f^a=fh,d^b=dfh,e^b=eh,f^b=fh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12065, #483589, #483590, … |
| maximal normal subgroup of | C2×#12065, #483589, #483590, … |
| maximal quotient of | #122961, #122964, #122967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5053);