Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
299 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 131 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)(3,5,8,4)(9,10,12,15,13,16,20,23)(11,17,19,24,18,14,21,22);(1,3,5,7,6,8,4,2)(9,11)(10,14,16,17)(12,19)(13,18)(15,22,23,24)(20,21);(1,4,6,5)(2,3,7,8)(11,18)(14,17)(19,21)(22,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105806, #105807, #105898, … |
| maximal normal subgroup of | #105806, #105807, #105898, … |
| maximal quotient of | #95240, #95242, #95244, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1070);