Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8608 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #380 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7)(17,18,20,24,21,23,29,31)(19,25,26,32,27,22,28,30);(1,3,5,12,6,13,4,11)(2,7,14,16,9,15,8,10);(17,19,21,27)(18,22,23,25)(20,28,29,26)(24,32,31,30)Polycyclic presentation:
2^8|a^2=f,b^2=dg,c^2=h,d^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93016, #93024, #122734, … |
| maximal normal subgroup of | #93016, #93024, #122734, … |
| maximal quotient of | #98819, #98821, #98833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1611);