Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
341 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2)(3,5)(4,6)(7,8)(9,10,14,19,15,20,13,16)(11,21,22,31,24,33,29,17)(12,25,28,34,26,30,27,18)(23,32,36,35,38,40,37,39);(1,3,4,7)(2,5,6,8)(10,16,20,19)(11,22,24,29)(12,26)(17,31)(18,25,34,30)(21,33)(23,37,38,36)(27,28);(2,6)(5,8)(9,11)(10,17)(12,23)(13,22)(14,29)(15,24)(16,21)(18,32)(19,33)(20,31)(25,39)(26,38)(27,36)(28,37)(30,35)(34,40);(9,12,13,27,15,26,14,28)(10,18,19,25,20,34,16,30)(11,23,22,36,24,38,29,37)(17,32,33,39,31,40,21,35)Polycyclic presentation:
2^8|a^2=g,b^2=ef,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #504676, #505192, #506179, … |
| maximal normal subgroup of | #504676, #505192, #506179, … |
| maximal quotient of | #315085, #315090, #315094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20798);