Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
763 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 763 conjugacy classes of subgroups — too many to draw. Only its 145 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:
(2,3)(5,6,10,13)(7,11,16,20)(8,17,12,9)(14,19,18,15);(1,2,4,3)(5,7,10,16)(6,11,13,20)(8,18,12,14)(9,19,17,15);(5,8,9,6,10,12,17,13)(7,14,15,11,16,18,19,20);(1,3,4,2)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=f,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #142612, #142614, #142740, … |
| maximal normal subgroup of | #142612, #142614, #142740, … |
| maximal quotient of | #297214, #297221, #297228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18729);