Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1861 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 1861 conjugacy classes of subgroups — too many to draw. Only its 303 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,7,14,12)(8,11,17,19,16,15,20,18);(5,7)(6,10)(8,15)(9,12)(11,16)(13,14)(17,18)(19,20);(2,4)(5,8)(6,11)(7,15)(9,17)(10,16)(12,18)(13,19)(14,20);(8,16)(11,15)(17,20)(18,19)Polycyclic presentation:
2^8|a^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #146589, #146590, #254467, … |
| maximal normal subgroup of | #146589, #146590, #254467, … |
| maximal quotient of | #184847, #184848, #184857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13594);