Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,20)(12,23)(13,17)(14,18)(15,19)(16,22)(21,24);(1,2,5,7)(3,6,4,8)(9,11,15,12)(10,16,19,24)(13,21,14,22)(17,20,18,23);(1,3,5,4)(2,6,7,8)(9,12,15,11)(10,16,19,24)(13,22,14,21)(17,20,18,23);(9,13,15,14)(10,17,19,18)(11,21,12,22)(16,20,24,23)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,e^b=eg,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #210266, #210274, #210282, … |
| maximal normal subgroup of | #210266, #210274, #210282, … |
| maximal quotient of | #165723, #165762, #165768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11510);