Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | Aut(G) order 4096 |
| 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 | C24 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
810 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 810 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(5,6)(7,13,15,10)(8,16,17,11)(9,12)(14,18)(19,20);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19);(2,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,17)(19,20);(5,8)(6,11)(7,14)(9,17)(10,18)(12,16)(13,19)(15,20)Polycyclic presentation:
2^8|a^2=f,b^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,e^c=eh,f^d=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #501085, #501088, #501091, … |
| maximal normal subgroup of | #501085, #501088, #501091, … |
| maximal quotient of | #267373, #267377, #267381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17394);