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 | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,22)(21,24);(10,15)(11,17)(13,23)(16,19)(18,24)(20,22);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,18,21,16)(13,15,23,17)(19,20,24,22);(9,12)(10,16)(11,18)(13,22)(14,21)(15,19)(17,24)(20,23)Polycyclic presentation:
2^8|a^2=fh,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25984, #6438157, #6438158, … |
| maximal normal subgroup of | C2×#25984, #6438157, #6438158, … |
| maximal quotient of | #305244, #305245, #305246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18972);