Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #466 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,27)(19,24,30,40)(20,31,34,45)(21,26,22,36)(25,32,41,33)(28,42,44,47)(29,38,35,39)(37,46,48,43);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,28)(21,35)(22,29)(23,30)(25,37)(26,39)(27,40)(31,42)(32,46)(33,43)(34,44)(36,38)(41,48)(45,47);(17,20,21,32,23,34,22,33)(18,25,36,45,27,41,26,31)(19,28,35,46,30,44,29,43)(24,37,38,47,40,48,39,42);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,32,34,33)(24,38,40,39)(25,31,41,45)(28,43,44,46)(37,47,48,42)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=gh,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131761, #131766, #489636, … |
| maximal normal subgroup of | #131761, #131766, #489636, … |
| maximal quotient of | #6386831, #6389319, #6389323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55368);