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 | #3287 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
328 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 328 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)(9,10,15,17)(11,18,21,22)(12,13,14,23)(16,24,20,19);(9,11,15,21)(10,16,17,20)(12,19,14,24)(13,22,23,18);(2,5)(3,8)(6,7)(9,12,15,14)(10,13,17,23)(11,19,21,24)(16,22,20,18);(9,13,15,23)(10,12,17,14)(11,20,21,16)(18,24,22,19)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=g,d^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 | #491779, #491787, #491794, … |
| maximal normal subgroup of | #491779, #491787, #491794, … |
| maximal quotient of | #220805, Q16×#103, #220824, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12974);