Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | Aut(G) order 8192 |
| 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 | #1445 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 109 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)(3,7,12,22)(4,14,16,29)(5,10)(6,9)(8,15,24,31)(11,26)(13,21,17,25)(18,19)(20,30)(23,28)(27,32);(1,3,5,17,6,12,18,13)(2,7,19,21,9,22,10,25)(4,11,15,30,16,27,31,28)(8,23,29,32,24,20,14,26);(1,4,5,15,6,16,18,31)(2,8,19,29,9,24,10,14)(3,11,17,30,12,27,13,28)(7,20,21,26,22,23,25,32)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11349, #358818, #361943, … |
| maximal normal subgroup of | C2×#11349, #358818, #361943, … |
| maximal quotient of | #121719, #121720, #121721, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4337);