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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #3484 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
849 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 849 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,13,15,10)(8,16)(9,12)(11,17)(14,19,20,18);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20);(2,4)(5,8)(6,11)(7,14)(9,17)(10,18)(12,16)(13,19)(15,20);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)Polycyclic presentation:
2^8|a^2=fh,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 | #501086, #501089, #501090, … |
| maximal normal subgroup of | #501086, #501089, #501090, … |
| maximal quotient of | #267372, #267376, #267380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17393);