Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3154 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
386 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)(4,5)(6,8)(9,10)(11,21)(12,23)(13,19)(14,18)(15,20)(16,22)(17,24);(2,5)(6,7)(9,10,14,19,15,20,13,18)(11,21,22,24,23,12,17,16);(9,11,14,22,15,23,13,17)(10,16,19,21,20,24,18,12);(1,3,4,8)(2,6,5,7)(9,12,14,16,15,21,13,24)(10,17,19,11,20,22,18,23)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=eh,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,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26275, #6444572, #6444574, … |
| maximal normal subgroup of | C2×#26275, #6444572, #6444574, … |
| maximal quotient of | #307419, #307420, #307423, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19263);