Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
597 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 183 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)(3,7)(4,6)(5,8)(9,10,13,18,14,17,26,31)(11,15,21,29,20,28,36,39)(12,22,23,32,24,16,25,30)(19,33,35,40,34,27,37,38);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,27)(17,28)(18,29)(22,33)(23,35)(24,34)(25,37)(26,36)(30,38)(31,39)(32,40);(3,8)(5,7)(9,12,14,24)(10,16,17,22)(11,19,20,34)(13,25,26,23)(15,27,28,33)(18,32,31,30)(21,37,36,35)(29,40,39,38);(2,6)(5,7)(11,20)(15,28)(19,34)(21,36)(27,33)(29,39)(35,37)(38,40)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #135555, #135556, #139801, … |
| maximal normal subgroup of | #135555, #135556, #139801, … |
| maximal quotient of | #215121, #215123, #215125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14790);