Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22);(1,2)(3,6)(4,5)(7,8);(2,5)(6,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=g,c^2=h,d^2=e,g^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 | #105762, #106095, #106104, … |
| maximal normal subgroup of | #105762, #106095, #106104, … |
| maximal quotient of | #216169, #216173, #216176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14867);