Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #397 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1439 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12)(4,17,21,9)(5,22)(6,10)(7,11)(8,14)(13,28,31,25)(15,29)(16,24)(18,23,30,32)(19,26,20,27);(1,3,5,14,7,15,6,16)(2,8,10,12,11,24,22,29)(4,18,19,13,21,30,20,31)(9,25,26,32,17,28,27,23);(1,4)(2,9)(3,13)(5,20)(6,19)(7,21)(8,23)(10,27)(11,17)(12,28)(14,18)(15,31)(16,30)(22,26)(24,32)(25,29)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^b=eh,d^c=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11132, #359281, #361803, … |
| maximal normal subgroup of | C2×#11132, #359281, #361803, … |
| maximal quotient of | #116989, #116990, #117015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4120);