Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9183 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
218 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,14,18,15,19,13,29)(11,20,24,31,23,35,22,16)(12,25,27,33,28,17,26,32)(21,34,38,30,37,40,36,39);(1,2,3,7,4,6,8,5)(9,11,15,23)(10,16)(12,21,28,37)(13,22,14,24)(17,30)(18,20)(19,31)(25,39)(26,36,27,38)(29,35)(32,40)(33,34);(9,12,15,28)(10,17,19,25)(11,21,23,37)(13,27,14,26)(16,30,31,39)(18,33,29,32)(20,34,35,40)(22,38,24,36)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=g,d^2=gh,f^2=g,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 | #131135, #131136, #131137, … |
| maximal normal subgroup of | #131135, #131136, #131137, … |
| maximal quotient of | #99824, #99826, #99828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2178);