Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #380 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #121 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
179 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 65 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:
(1,2,4,7,5,3,8,6)(9,10)(11,20)(12,25,27,32)(13,28)(14,18)(15,19)(16,22)(17,26,33,29)(21,30,37,40)(23,31)(24,35)(34,36,39,38);(9,11,14,23,15,24,13,22)(10,16,28,35,19,31,18,20)(12,21,29,38,27,37,26,36)(17,30,25,39,33,40,32,34);(1,3)(2,5)(4,6)(7,8)(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,32)(19,33)(20,34)(22,36)(23,38)(24,37)(25,28)(31,40)(35,39)Polycyclic presentation:
2^8|a^2=f,b^2=dg,d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=egh,f^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | #131668, #131669, #131700, … |
| maximal normal subgroup of | #131668, #131669, #131700, … |
| maximal quotient of | #98890, #98891, #98902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1679);