Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
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×C4⋊C8 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
227 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 95 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)(5,6,9,14,10,13,23,28,11,15,24,29,8,21,22,34)(7,16,19,31,20,26,33,35,17,30,32,36,18,12,25,27);(5,7,8,18,11,17,10,20)(6,12,13,16,15,26,21,30)(9,19,22,25,24,32,23,33)(14,27,28,31,29,35,34,36);(2,4)(7,17)(12,26)(16,30)(18,20)(19,32)(25,33)(27,35)(31,36)Polycyclic presentation:
2^8|a^2=f,b^2=d,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 | #93417, #93437, #93581, … |
| maximal normal subgroup of | #93417, #93437, #93581, … |
| maximal quotient of | #134223, #134251, #134277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5777);