Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8608 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #380 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
199 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 83 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:
(2,5)(3,7)(6,8)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22);(1,2,3,5,4,6,7,8);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)Polycyclic presentation:
2^8|a^2=f,b^2=d,c^2=h,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93014, #93028, #122814, … |
| maximal normal subgroup of | #93014, #93028, #122814, … |
| maximal quotient of | #98817, #98821, #98829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1603);