Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
785 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 785 conjugacy classes of subgroups — too many to draw. Only its 187 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:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(1,3)(2,5)(4,8)(6,7)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23);(3,8)(5,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(2,6)(5,7)Polycyclic presentation:
2^8|a^2=g,b^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #107154, #107174, #110092, … |
| maximal normal subgroup of | #107154, #107174, #110092, … |
| maximal quotient of | #213493, #213495, #213498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14735);