Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = C23×#7684106 (order 4096) · Inn(G) = C22⋊C4 · Out(G) = C25×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
241 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 97 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:
(2,6)(3,4)(7,8)(9,10,13,19,15,20,14,18)(11,21,22,24,23,12,17,16);(1,2,4,7,5,8,3,6)(9,11,14,17,15,23,13,22)(10,16,18,12,20,24,19,21);(9,12,13,16,15,21,14,24)(10,17,19,11,20,22,18,23)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,f^2=g,b^a=bd,d^a=dg,f^a=fgh,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #128662, #128663, #128686, … |
| maximal normal subgroup of | #128662, #128663, #128686, … |
| maximal quotient of | #96774, #96784, #96787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1239);