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) = C22×C4 |
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 | |
232 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 232 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:
(1,2,7,11,8,12,6,24)(3,5,10,22,16,23,15,13)(4,17,20,26,21,31,19,9)(14,28,25,32,30,27,29,18);(1,3,13,2)(4,14,27,9)(5,11,6,15)(7,10,23,24)(8,16,22,12)(17,19,29,18)(20,25,32,31)(21,30,28,26);(1,4,7,20,8,21,6,19)(2,9,24,31,12,26,11,17)(3,14,10,25,16,30,15,29)(5,18,13,27,23,32,22,28)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=eh,e^2=h,f^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 | #138938, #138939, #138948, … |
| maximal normal subgroup of | #138938, #138939, #138948, … |
| maximal quotient of | #140617, #140635, #140648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6222);