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) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3041 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
383 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)(9,10,14,18)(11,20,13,17)(12,22,24,21)(15,23,16,19);(2,5)(3,8)(9,11,14,13)(10,17,18,20)(12,23,24,19)(15,21,16,22);(9,12)(10,13)(11,18)(14,24)(15,17)(16,20)(19,21)(22,23);(1,3,4,8)(2,6,5,7)(9,13)(10,12)(11,14)(15,21)(16,22)(17,19)(18,24)(20,23)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=eh,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 | #6670990, #6670997, #6671002, … |
| maximal normal subgroup of | #6670990, #6670997, #6671002, … |
| maximal quotient of | #340444, #340445, #340456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23931);