Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#72317393 (order 4096) · Inn(G) = C23 · Out(G) = C22×C4×#120 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C23 |
| commutator, socle | C22 | C22×C16 |
| prime radical | C24 | C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,7,12,8,13,15,24,9,14,16,25,18,26,27,34,10,11,17,21,19,22,28,31,20,23,29,32,30,33,35,36);(2,4);(6,11)(12,21)(13,22)(14,23)(24,31)(25,32)(26,33)(34,36)Polycyclic presentation:
2^8|a^2=e,e^2=f,f^2=g,g^2=h,b^a=bd,c^a=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | #94061, #94062, #94123, … |
| maximal normal subgroup of | #94061, #94062, #94123, … |
| maximal quotient of | #149518, #149600, #149654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6541);