Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18866 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3497 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
561 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(11,18)(12,13)(15,22)(16,20)(19,21)(23,24);(2,6)(3,4)(7,8)(9,10)(11,15)(12,16)(13,20)(14,17)(18,23)(19,24)(21,22);(9,11,14,19)(10,15,17,24)(12,18,13,21)(16,23,20,22);(1,3,5,4)(2,6,8,7)(9,12,14,13)(10,16,17,20)(11,18,19,21)(15,23,24,22)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #250559, #250561, #250575, … |
| maximal normal subgroup of | #250559, #250561, #250575, … |
| maximal quotient of | #167061, #167065, #167072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11901);