Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2308 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
631 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 631 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13,17,8)(4,10,5,19)(6,12)(7,11)(9,18,27,20)(14,30)(15,24)(16,29,21,25)(22,28)(23,31)(26,32);(1,3)(2,8)(4,14)(5,15)(6,21)(7,17)(9,26)(10,24)(11,13)(12,29)(16,22)(18,32)(19,30)(20,31)(23,27)(25,28);(1,4,7,5)(2,9,11,27)(3,14,17,15)(6,20,22,18)(8,23,13,26)(10,12,19,28)(16,32,21,31)(24,25,30,29);(1,5,7,4)(2,10,11,19)(3,15,17,14)(6,20,22,18)(8,24,13,30)(9,12,27,28)(16,32,21,31)(23,25,26,29)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=gh,g^d=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #151821, #151826, #151828, … |
| maximal normal subgroup of | #151821, #151826, #151828, … |
| maximal quotient of | #213769, #213773, #213774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14773);