Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,20)(13,19)(15,17)(16,24)(18,22)(21,23);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,21,23,20)(13,19,15,17)(16,22,18,24);(9,12)(10,16)(11,18)(13,22)(14,23)(15,24)(17,21)(19,20);(2,6)(5,7)(9,13,14,15)(10,17,11,19)(12,22,23,24)(16,21,18,20)Polycyclic presentation:
2^8|b^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23956, #6451761, #6451764, … |
| maximal normal subgroup of | C2×#23956, #6451761, #6451764, … |
| maximal quotient of | #251401, #251403, #251404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16944);