Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| 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 | #3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
451 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,14)(13,23)(15,19)(16,20)(17,18)(22,24);(2,5)(6,8)(9,11,15,22,16,23,14,18)(10,17,12,13,20,24,19,21);(1,3,4,7)(2,6,5,8)(9,12,14,10,16,19,15,20)(11,17,18,21,23,24,22,13);(2,5)(6,8)(9,13,15,24,16,21,14,17)(10,18,12,11,20,22,19,23)Polycyclic presentation:
2^8|b^2=f,c^2=e,d^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27286, #6440834, #6440840, … |
| maximal normal subgroup of | C2×#27286, #6440834, #6440840, … |
| maximal quotient of | #313577, #313580, #313589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20274);