Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,22)(18,24)(21,23);(3,7)(5,8)(9,11)(10,14)(12,21)(13,18)(15,23)(16,24)(17,20)(19,22);(1,2)(3,8)(4,6)(5,7);(10,15)(11,18)(16,20)(19,21)Polycyclic presentation:
2^8|e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18618, #6348203, #6348263, … |
| maximal normal subgroup of | C2×#18618, #6348203, #6348263, … |
| maximal quotient of | #166737, #166754, #166771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11606);