Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| 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 | C42 | C24 |
| prime radical | #547 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7)(3,5,8,6)(10,14)(12,21)(15,22)(16,17)(18,23)(19,24);(9,10,13,17)(11,18,20,15)(12,16,21,14)(19,22,24,23);(2,5)(3,8)(6,7)(9,11,13,20)(10,15,17,18)(12,19,21,24)(14,22,16,23);(10,16)(12,21)(14,17)(15,23)(18,22)(19,24)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=de,e^b=eg,f^c=fh,e^d=eg43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19894, #6358265, #6358345, … |
| maximal normal subgroup of | C2×#19894, #6358265, #6358345, … |
| maximal quotient of | #220770, #220781, #220948, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12882);