Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#397 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,10)(3,13,15,8,17,24,16,25)(4,18,20,26,21,27,22,9)(14,29,31,32,19,23,30,28);(1,3)(2,8)(4,19)(5,15)(6,16)(7,17)(9,23)(10,24)(11,13)(12,25)(14,21)(18,32)(20,31)(22,30)(26,29)(27,28);(1,4,5,20,7,21,6,22)(2,9,10,18,12,26,11,27)(3,14,16,31,17,19,15,30)(8,23,13,28,25,29,24,32);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,20,21,22)(8,24,25,13)(9,18,26,27)(14,30,19,31)(23,32,29,28)Polycyclic presentation:
2^8|a^2=f,c^2=e,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #359281, #376658, #378906, … |
| maximal normal subgroup of | #359281, #376658, #378906, … |
| maximal quotient of | #242952, #242955, #242957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16640);