Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) = C25×#442 (order 2048) · Inn(G) = C2×D8 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C2×D8 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
119 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 119 conjugacy classes of subgroups — too many to draw. Only its 57 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)(11,14)(12,21)(13,15)(16,18)(17,19)(20,22)(23,24);(10,14)(11,18)(12,19)(15,22)(16,17)(20,24)(21,23);(1,3,4,8,5,2,6,7)(10,15)(14,22)(16,23)(17,21)Polycyclic presentation:
2^8|a^2=e,c^2=e,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=ch,d^a=df,f^a=fh,d^b=df,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #122961, #122964, #122967, … |
| maximal normal subgroup of | #122961, #122964, #122967, … |
| maximal quotient of | #122944, #122950, #122957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5031);