Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| 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 | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
468 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 131 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:
(2,5)(3,7)(6,8)(9,10,15,20)(11,17,22,23)(12,16,21,24)(13,19,14,18);(1,2,4,6)(3,8,7,5)(9,11,15,22)(10,16,20,24)(12,13,21,14)(17,18,23,19);(9,12)(10,17)(11,13)(14,22)(15,21)(16,18)(19,24)(20,23);(9,13,15,14)(10,18,20,19)(11,21,22,12)(16,23,24,17)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,e^b=eg,f^b=fh,e^c=eh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #270639, #270640, #270679, … |
| maximal normal subgroup of | #270639, #270640, #270679, … |
| maximal quotient of | #166081, #166085, #166091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11788);