Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
241 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 241 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)(19,27,29,31)(20,30,23,24)(21,25)(22,26)(28,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19,21,30,22,29,32,24)(18,23,25,31,26,20,28,27);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,22,23)(18,24,26,30)(19,28,29,25)(21,27,32,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132957, #132958, #132959, … |
| maximal normal subgroup of | #132957, #132958, #132959, … |
| maximal quotient of | #99548, #99550, #99556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2022);