Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3455 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
529 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 529 conjugacy classes of subgroups — too many to draw. Only its 183 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10,13,17,14,12,21,16)(11,15,20,23,19,18,24,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,20)(14,19)(16,22)(17,23)(21,24);(2,5)(6,8)(9,12)(10,14)(11,18)(13,16)(15,19)(17,21)(20,22)(23,24);(1,3,4,7)(2,6,5,8)(11,19)(15,18)(20,24)(22,23)Polycyclic presentation:
2^8|a^2=g,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #253339, #253340, #253345, … |
| maximal normal subgroup of | #253339, #253340, #253345, … |
| maximal quotient of | #216352, #216353, #216354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14889);