Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| 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 | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | #3150 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
251 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 251 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,6,8,5,3,4,7)(9,10)(11,20,23,31)(12,24)(13,18)(14,28)(15,19)(16,22,32,27)(17,25)(21,36,39,30)(26,33)(29,34)(35,37,40,38);(1,3,5,2)(4,7,6,8)(9,11,14,22,15,23,13,27)(10,16,28,20,19,32,18,31)(12,21,29,37,26,39,25,38)(17,30,24,35,34,36,33,40);(4,6)(7,8)(9,12)(10,17)(11,21)(13,25)(14,29)(15,26)(16,30)(18,33)(19,34)(20,35)(22,37)(23,39)(24,28)(27,38)(31,40)(32,36)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152184, #152476, C2×#9007, … |
| maximal normal subgroup of | #152184, #152476, C2×#9007, … |
| maximal quotient of | #99537, #99539, #99543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1995);