Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
621 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 621 conjugacy classes of subgroups — too many to draw. Only its 135 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)(3,4)(9,10,14,20,15,18,13,19)(11,21,24,32,25,16,23,33)(12,26,29,35,30,17,28,36)(22,31,38,40,39,37,27,34);(2,4)(5,6)(7,8)(9,11,15,25)(10,16,18,21)(12,27,30,38)(13,24,14,23)(17,34,26,40)(19,33,20,32)(22,29,39,28)(31,36,37,35);(6,8)(9,12,15,30)(10,17,18,26)(11,22,25,39)(13,29,14,28)(16,31,21,37)(19,36,20,35)(23,38,24,27)(32,40,33,34)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93899, #137376, #137377, … |
| maximal normal subgroup of | #93899, #137376, #137377, … |
| maximal quotient of | #111223, #111277, #111281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4017);