Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
566 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(5,8)(9,10,15,12)(11,19,22,24)(13,18,14,17)(16,23,20,21);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,19)(18,24);(1,3)(2,5)(4,7)(6,8)(9,12,15,10)(11,20,22,16)(13,17,14,18)(19,23,24,21);(9,13,15,14)(10,17,12,18)(11,21,22,23)(16,19,20,24)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #249888, #249899, #249922, … |
| maximal normal subgroup of | #249888, #249899, #249922, … |
| maximal quotient of | #166998, #167020, #167061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11247);