Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3334 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
345 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 111 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)(3,5)(4,7)(6,8)(9,10,14,12)(11,17,20,24)(13,22,23,16)(15,18,21,19);(9,11,14,20)(10,15,12,21)(13,18,23,19)(16,24,22,17);(2,5)(3,8)(6,7)(9,12,14,10)(11,15,20,21)(13,22,23,16)(17,18,24,19);(11,18)(13,23)(15,24)(16,22)(17,21)(19,20)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=de,e^b=eg,f^c=fh,e^d=eg43 conjugacy classes — view the full character table.
| maximal subgroup of | #491840, #491842, #491856, … |
| maximal normal subgroup of | #491840, #491842, #491856, … |
| maximal quotient of | #220770, #220966, #220972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12875);