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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
424 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,21)(13,17)(14,18)(15,24)(16,19)(22,23);(1,3,5,2)(4,7,8,6)(9,11)(10,17)(12,19)(13,14)(15,21)(16,23)(18,20)(22,24);(1,4,5,8)(2,6,3,7)(9,12)(10,11)(13,18)(14,22)(15,17)(16,20)(19,23)(21,24);(2,3)(6,7)(9,13,14,11)(10,12,18,22)(15,21,16,23)(17,24,20,19)Polycyclic presentation:
2^8|b^2=eh,c^2=eh,d^2=h,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 | #6669048, #6669052, #6669060, … |
| maximal normal subgroup of | #6669048, #6669052, #6669060, … |
| maximal quotient of | #337676, #337677, #337680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23760);