Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | D4×C4⋊C4 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,25)(3,14,16,9,18,27,17,28)(4,19,15,29,21,26,20,10)(5,11,22,31,24,32,23,30);(1,3)(2,9)(4,5)(6,17)(7,16)(8,18)(10,11)(12,27)(13,28)(14,25)(15,22)(19,31)(20,23)(21,24)(26,30)(29,32);(1,4,6,20,8,21,7,15)(2,10,12,19,13,29,25,26)(3,5,17,23,18,24,16,22)(9,11,27,31,28,32,14,30);(1,5)(2,11)(3,15)(4,17)(6,23)(7,22)(8,24)(9,26)(10,27)(12,31)(13,32)(14,29)(16,21)(18,20)(19,28)(25,30)Polycyclic presentation:
2^8|a^2=g,c^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136326, #136328, #136363, … |
| maximal normal subgroup of | #136326, #136328, #136363, … |
| maximal quotient of | #225494, #225495, #225496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15682);