Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C22×M16 | C22 |
| socle | C22 | #323 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 55 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 — 2 generators on 36 points:
(5,6,9,14,10,13,24,29,11,15,25,31,8,21,22,30)(7,16,18,32,19,28,34,36,20,33,23,35,17,12,26,27);(1,2,3,4)(5,7,8,17,11,20,10,19)(6,12,13,16,15,28,21,33)(9,23,22,34,25,18,24,26)(14,27,29,32,31,36,30,35)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=g,e^2=h,g^2=h,b^a=bc,c^a=cf,e^a=eh,f^a=fh,d^b=dfh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93898, #93899, #93906, … |
| maximal normal subgroup of | #93898, #93899, #93906, … |
| maximal quotient of | #92968, #92969, #92972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 182);