Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #391 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1441 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
320 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12)(4,16,19,31)(5,20)(6,10)(7,11)(8,14)(9,18,26,17)(13,22,29,25)(15,23)(21,24)(27,28,30,32);(1,3,5,14,7,15,6,21)(2,8,10,12,11,24,20,23)(4,13,17,28,19,29,18,32)(9,25,16,30,26,22,31,27);(1,4,5,17,7,19,6,18)(2,9,10,16,11,26,20,31)(3,13,14,28,15,29,21,32)(8,22,12,27,24,25,23,30)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131436, #131437, #144754, … |
| maximal normal subgroup of | #131436, #131437, #144754, … |
| maximal quotient of | #120168, #120186, #120188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4222);