Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | C2×C4×D4 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 169 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,10,12,16)(7,13)(11,18,19,14)(15,20);(5,6,8,16,9,12,17,10)(7,14,15,11,13,18,20,19);(1,2,3,4);(5,7,8,15,9,13,17,20)(6,11,10,14,12,19,16,18)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136500, #136508, #136516, … |
| maximal normal subgroup of | #136500, #136508, #136516, … |
| maximal quotient of | #231411, #231412, #231413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16168);