Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3486 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
680 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 137 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:
(5,6,12,20,10,15,11,14)(7,13,18,17,9,8,16,19);(1,2)(3,4)(5,7,10,9)(6,13,15,8)(11,18,12,16)(14,17,20,19);(2,4)(5,8)(6,7)(9,15)(10,13)(11,19)(12,17)(14,18)(16,20);(5,9,10,7)(6,8,15,13)(11,16,12,18)(14,19,20,17)Polycyclic presentation:
2^8|a^2=fh,b^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,e^c=eh,f^c=fh,f^d=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #501122, #501125, #501126, … |
| maximal normal subgroup of | #501122, #501125, #501126, … |
| maximal quotient of | #267474, #267482, #267490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17407);