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 |
| prime radical | #3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
447 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)(9,10,15,16,13,18,14,19)(11,20,22,23,17,12,21,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,17)(14,21)(15,22)(16,24)(18,20)(19,23);(10,16)(11,21)(14,15)(17,22)(18,19)(23,24);(2,6)(3,8)(9,12)(10,17)(11,18)(13,20)(14,24)(15,23)(16,22)(19,21)Polycyclic presentation:
2^8|a^2=fh,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 | #6667749, #6667753, #6667755, … |
| maximal normal subgroup of | #6667749, #6667753, #6667755, … |
| maximal quotient of | #342043, #342044, #342045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24038);