Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| 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 | #3559 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
472 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 113 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)(9,10,14,12)(11,17,19,24)(13,21,23,16)(15,22,18,20);(3,7)(5,8)(9,11)(10,15)(12,18)(13,22)(14,19)(16,24)(17,21)(20,23);(1,2)(3,8)(4,6)(5,7)(9,12,14,10)(11,18,19,15)(13,16,23,21)(17,22,24,20);(11,19)(15,18)(17,24)(20,22)Polycyclic presentation:
2^8|a^2=g,c^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | #250872, #250874, #250883, … |
| maximal normal subgroup of | #250872, #250874, #250883, … |
| maximal quotient of | #166292, #166355, #166420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12005);