Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C24×#36614 (order 4096) · Inn(G) = C23 · Out(G) = C23×#525 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C23 |
| commutator | C22 | C2×C4×C8 |
| socle | C23 | C22×C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
237 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 237 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,12,16,13,11,18,19,14,17,21,24,22,20,15,23);(2,4)(5,6,7,8);(9,11,12,19,13,17,18,24,14,20,21,23,22,10,15,16)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,e^2=g,g^2=h,b^a=bd,c^a=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | #152814, C2×#12612, #492964, … |
| maximal normal subgroup of | #152814, C2×#12612, #492964, … |
| maximal quotient of | #128247, #128264, #128282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5600);