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 | D4×#41 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
603 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 603 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10,14,19)(11,20,23,24)(12,18,13,17)(15,21,16,22);(2,5)(3,8)(6,7)(9,11)(10,15)(12,21)(13,22)(14,23)(16,19)(17,24)(18,20);(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24);(9,12,14,13)(10,17,19,18)(11,21,23,22)(15,24,16,20)Polycyclic presentation:
2^8|a^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 | #270631, #270632, #270655, … |
| maximal normal subgroup of | #270631, #270632, #270655, … |
| maximal quotient of | #165914, #165919, #165924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11775);