Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) = C22×#72219693 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator, prime radical | C23 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C22⋊C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
209 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 95 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:
(1,2,4,3,5,6,8,7)(9,10,12,15)(11,17,18,14)(13,21,20,16)(19,22,24,23);(9,11,13,19,12,18,20,24)(10,14,16,22,15,17,21,23);(1,3,4,6,5,7,8,2)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=eg,e^2=h,f^2=g,b^a=bd,c^a=ch,f^a=fg,d^b=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #140162, C2×#8167, #182100, … |
| maximal normal subgroup of | #140162, C2×#8167, #182100, … |
| maximal quotient of | #96677, #96679, #96680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1155);