Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,19,15,12,13,18)(11,16,24,17,22,23,21,20);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,23)(13,21)(14,24)(15,22)(17,19)(18,20);(9,12,15,10)(11,20,22,17)(13,18,14,19)(16,24,23,21);(2,6)(3,8)(9,11)(10,17)(12,20)(13,24)(14,21)(15,22)(16,19)(18,23)Polycyclic presentation:
2^8|a^2=fh,c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6667750, #6667752, #6667756, … |
| maximal normal subgroup of | #6667750, #6667752, #6667756, … |
| maximal quotient of | #342047, #342048, #342049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24039);