Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1964 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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)(3,5,13,20)(4,14)(6,21)(7,10)(8,17)(9,15)(11,23,26,32)(12,27,24,19)(16,30,31,25)(18,28)(22,29);(1,3,5,10,7,13,20,2)(4,11,16,22,18,26,31,8)(6,12,19,21,15,24,27,9)(14,17,25,23,28,29,30,32);(1,4,15,29,7,18,6,17)(2,8,21,28,10,22,9,14)(3,11,24,30,13,26,12,25)(5,16,27,32,20,31,19,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=eh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,f^b=fg,g^b=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #138940, #138941, #138946, … |
| maximal normal subgroup of | #138940, #138941, #138946, … |
| maximal quotient of | #140617, #140634, #140647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6219);