Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3283 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
336 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 336 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:
(1,2,4,7)(3,5,8,6)(10,15)(12,13)(16,22)(17,21)(18,19)(23,24);(9,10)(11,16)(12,15)(13,21)(14,17)(18,22)(19,24)(20,23);(2,5)(3,8)(6,7)(9,11,14,20)(10,16,17,23)(12,18,13,19)(15,22,21,24);(9,12)(11,18)(13,14)(15,21)(19,20)(22,24)Polycyclic presentation:
2^8|a^2=h,c^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,d^b=de,e^b=eg,f^c=fh,e^d=eg49 conjugacy classes — view the full character table.
| maximal subgroup of | #491777, #491785, #491792, … |
| maximal normal subgroup of | #491777, #491785, #491792, … |
| maximal quotient of | #220802, #220810, D8×#103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12967);