Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| 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 | #469 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
800 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)(19,24)(20,31)(21,26)(22,36)(23,27)(25,34)(28,42)(29,38)(30,40)(32,45)(33,41)(35,39)(37,44)(43,48)(46,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,23,30)(18,24,27,40)(20,28,32,43)(21,35,22,29)(25,37,41,47)(26,39,36,38)(31,42,45,48)(33,46,34,44);(17,20,22,34,23,32,21,33)(18,25,26,31,27,41,36,45)(19,28,29,44,30,43,35,46)(24,37,39,42,40,47,38,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(20,32)(25,41)(28,43)(31,45)(33,34)(37,47)(42,48)(44,46);(2,8)(6,9)(7,16)(14,15)(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,33,32,34)(24,38,40,39)(25,31,41,45)(28,44,43,46)(37,48,47,42)Polycyclic presentation:
2^8|b^2=fh,c^2=g,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=eh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #486085, C2×#62391, #10578435, … |
| maximal normal subgroup of | #486085, C2×#62391, #10578435, … |
| maximal quotient of | #6386837, #6389337, #6389362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55379);