Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18866 |
| automorphism | Aut(G) order 8192 |
| 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 | #3499 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
548 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 548 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)(9,10,15,18)(11,19,16,24)(12,20,22,21)(13,17,14,23);(2,6)(3,4)(7,8)(9,11,15,16)(10,12,18,22)(13,21,14,20)(17,24,23,19);(9,12)(10,16)(11,18)(13,19)(14,24)(15,22)(17,21)(20,23);(1,3,5,4)(2,6,8,7)(9,13,15,14)(10,17,18,23)(11,20,16,21)(12,19,22,24)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=g,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 | #245388, #245392, #245396, … |
| maximal normal subgroup of | #245388, #245392, #245396, … |
| maximal quotient of | #167058, #167068, #167078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11871);