Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 117 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 — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,16)(12,24)(13,29)(14,18)(15,19)(17,26)(20,33)(21,38)(22,31)(23,32)(25,39)(27,34)(28,35)(30,36)(37,40);(2,5)(6,7)(9,11,14,22,15,23,13,21)(10,16,18,31,19,32,29,38)(12,25,27,37,28,36,26,20)(17,33,24,39,34,40,35,30);(9,12,15,28)(10,17,19,34)(11,20,23,37)(13,27,14,26)(16,30,32,39)(18,35,29,24)(21,25,22,36)(31,40,38,33);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139679, #139681, #143111, … |
| maximal normal subgroup of | #139679, #139681, #143111, … |
| maximal quotient of | #317535, #317536, #317539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21207);