Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9183 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
229 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,11,12,7,14,15,13,16)(17,18,21,25,22,24,31,19)(20,27,29,32,28,23,30,26);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19,21,24,22,25,31,18)(20,26,30,27,28,32,29,23);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,22,28)(18,23,24,27)(19,26,25,32)(21,30,31,29)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #132989, #132992, #132994, … |
| maximal normal subgroup of | #132989, #132992, #132994, … |
| maximal quotient of | #99821, #99822, #99835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2180);