Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×C4×C22⋊C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,17,23)(3,11,14,25,13,7,18,20)(4,8,16,22,15,21,28,31)(12,24,27,32,26,19,29,30);(1,3)(2,7,10,11)(4,12)(5,14)(6,13)(8,19,21,24)(9,20,23,25)(15,26)(16,27)(17,18)(22,30,31,32)(28,29);(1,4,5,16,6,15,17,28)(2,8,9,22,10,21,23,31)(3,12,14,27,13,26,18,29)(7,19,20,30,11,24,25,32);(4,15)(8,21)(12,26)(16,28)(19,24)(22,31)(27,29)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,f^2=h,b^a=be,g^a=gh,e^b=eh,d^c=dh82 conjugacy classes — view the full character table.
| maximal subgroup of | #147327, #147328, #147503, … |
| maximal normal subgroup of | #147327, #147328, #147503, … |
| maximal quotient of | #182036, #182037, #182040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13389);