Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | #525 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
244 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 83 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:
(1,2)(3,4)(5,6,9,14,10,13,23,26,11,15,24,29,8,22,17,32)(7,16,20,31,21,27,34,35,18,30,33,36,19,12,25,28);(5,7,17,25,8,19,24,33,11,18,23,34,10,21,9,20)(6,12,26,31,22,30,14,28,15,27,32,36,13,16,29,35);(2,4)(7,18)(12,27)(16,30)(19,21)(20,33)(25,34)(28,35)(31,36)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #129597, #129598, #130254, … |
| maximal normal subgroup of | #129597, #129598, #130254, … |
| maximal quotient of | #136578, #136586, #136594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5972);