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) = C22×C4 |
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 | C8×D4 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
215 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 215 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:
(2,4)(5,6,11,17,12,16,27,33,9,15,19,14,10,28,8,25)(7,18,23,32,24,34,20,35,21,13,29,30,22,31,26,36);(1,2,3,4)(5,7,19,29,12,24,8,26,9,21,11,23,10,22,27,20)(6,13,14,32,16,31,25,35,15,18,17,30,28,34,33,36);(5,8,10,19,9,27,12,11)(6,14,16,25,15,17,28,33)(7,20,22,23,21,26,24,29)(13,30,31,36,18,32,34,35)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=e,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #93891, #136680, #136684, … |
| maximal normal subgroup of | #93891, #136680, #136684, … |
| maximal quotient of | #140510, #140516, #140538, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6184);