Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
387 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)(3,4)(5,7)(6,8)(9,10)(11,19)(12,17)(13,15)(14,18)(16,23)(20,22)(21,24);(1,3)(2,6)(4,7)(5,8)(9,10,12,17)(11,20,21,16)(13,18,14,15)(19,23,24,22);(1,4,5,6)(2,3,7,8)(10,15)(13,14)(16,24)(17,18)(19,20)(22,23);(2,7)(3,8)(9,11,12,21)(10,16,17,20)(13,23,14,22)(15,24,18,19)Polycyclic presentation:
2^8|b^2=h,c^2=eh,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #6669052, #6669055, #6669057, … |
| maximal normal subgroup of | #6669052, #6669055, #6669057, … |
| maximal quotient of | #335628, #335629, #335632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23616);