Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #380 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,11,16,10,14)(7,17,19,12,21,27,20,28)(8,22,23,29,24,13,25,30)(18,32,33,36,34,31,35,26);(1,2,3,4)(5,7,10,20,11,21,9,19)(6,12,15,27,16,28,14,17)(8,18,25,35,24,34,23,33)(13,26,30,32,22,36,29,31);(5,8,9,23,11,24,10,25)(6,13,14,29,16,22,15,30)(7,18,19,33,21,34,20,35)(12,26,17,31,28,36,27,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129950, #129951, #138014, … |
| maximal normal subgroup of | #129950, #129951, #138014, … |
| maximal quotient of | #108548, #108549, #108550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3816);