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 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 312 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:
(1,2,6,10,7,11,5,19)(3,12,16,22,15,28,14,8)(4,9,17,26,18,25,20,24)(13,29,30,23,31,27,32,21);(1,3,5,14,7,15,6,16)(2,8,10,12,11,22,19,28)(4,13,17,30,18,31,20,32)(9,23,24,29,25,21,26,27);(1,4,5,17,7,18,6,20)(2,9,10,24,11,25,19,26)(3,13,14,30,15,31,16,32)(8,21,12,27,22,23,28,29)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131432, #131433, #144792, … |
| maximal normal subgroup of | #131432, #131433, #144792, … |
| maximal quotient of | #120174, #120198, #120199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4234);