Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3153 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
468 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,21)(12,25)(13,20)(14,19)(15,18)(16,23)(17,26)(22,30)(24,33)(27,34)(28,31)(29,32)(35,37)(36,39)(38,40);(3,7)(5,8)(9,11)(10,16)(12,22)(13,28)(14,23)(15,24)(17,30)(18,31)(19,33)(20,21)(25,39)(26,37)(27,38)(29,36)(32,40)(34,35);(1,2)(3,8)(4,6)(5,7)(9,12,15,27)(10,17,18,32)(11,22,24,38)(13,26,14,29)(16,30,31,40)(19,25,20,34)(21,35,33,39)(23,36,28,37);(10,18)(16,31)(17,32)(19,20)(21,33)(25,34)(30,40)(35,39)Polycyclic presentation:
2^8|c^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152611, #245960, #245961, … |
| maximal normal subgroup of | #152611, #245960, #245961, … |
| maximal quotient of | #165875, #165895, #165917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11982);