Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #392 |
| 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 | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,17,23,16,24,15,8)(4,9,19,27,20,28,21,26)(14,29,18,22,30,25,31,32);(1,3,5,15,7,16,6,17)(2,8,10,13,12,23,11,24)(4,18,19,30,20,31,21,14)(9,25,26,22,28,29,27,32);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,22)(10,27)(11,26)(12,28)(13,25)(15,31)(16,30)(17,18)(23,32)(24,29)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=eh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11131, #359071, #361805, … |
| maximal normal subgroup of | C2×#11131, #359071, #361805, … |
| maximal quotient of | #116987, #116988, #117017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4119);