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 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,13)(12,20)(14,16)(15,18)(17,23)(19,22)(21,24);(1,3,5,8)(2,6,7,4)(9,11,13,10)(12,17,21,19)(14,16,15,18)(20,22,24,23);(10,16)(11,18)(14,15)(17,24)(19,20)(22,23);(1,4,5,6)(2,3,7,8)(9,12,13,21)(10,17,11,19)(14,23,15,22)(16,24,18,20)Polycyclic presentation:
2^8|a^2=eh,b^2=e,d^2=e,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 | #6672286, #6672287, #6672292, … |
| maximal normal subgroup of | #6672286, #6672287, #6672292, … |
| maximal quotient of | #341461, #341462, #341463, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23969);