Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8610 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
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 | |
171 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18,20,24,21,23,29,31)(19,25,26,32,27,22,28,30);(1,3,4,12,6,11,5,13)(2,7,8,10,9,15,14,16);(3,11)(7,15)(10,16)(12,13)(17,19,21,27)(18,22,23,25)(20,28,29,26)(24,32,31,30)Polycyclic presentation:
2^8|a^2=f,b^2=d,c^2=h,d^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130468, #130469, #131077, … |
| maximal normal subgroup of | #130468, #130469, #131077, … |
| maximal quotient of | #98825, #98827, #98829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1613);