Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| 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 | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
379 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10,14,16,15,11,13,18)(12,20,22,17,23,24,21,19);(2,5)(6,7)(10,16)(11,18)(13,14)(17,20)(19,24)(21,22);(9,11)(10,15)(12,19)(13,18)(14,16)(17,23)(20,21)(22,24);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,19)(13,22)(14,21)(15,23)(16,20)(18,24)Polycyclic presentation:
2^8|a^2=eh,d^2=ef,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh,e^d=eh,f^d=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26921, #6439732, #6439733, … |
| maximal normal subgroup of | C2×#26921, #6439732, #6439733, … |
| maximal quotient of | #312028, #312029, #312030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19909);