Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) = C24×#63011 (order 4096) · Inn(G) = #121 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #390 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
233 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 73 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,7,5,3,4,8)(9,10,14,18,15,17,13,11)(12,20,22,24,23,19,21,16);(1,3,6,8,5,2,4,7)(9,11,14,17,15,18,13,10)(12,20,22,16,23,19,21,24);(1,4,5,6)(2,7,3,8)(9,12,14,22,15,23,13,21)(10,16,11,19,17,24,18,20)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=eg,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=dg,f^b=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9672, #204435, #204436, … |
| maximal normal subgroup of | C2×#9672, #204435, #204436, … |
| maximal quotient of | #102013, #102014, #102015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2660);