Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| 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 | #535 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,30)(21,26)(22,36)(23,27)(25,28)(29,38)(31,40)(32,44)(33,37)(34,41)(35,42)(39,43)(45,48)(46,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,23,22)(18,24,27,36)(20,31,33,28)(21,34,35,29)(25,30,40,37)(26,41,42,38)(32,43,46,45)(39,47,48,44);(3,10)(6,9)(11,16)(13,15)(17,20,23,33)(18,25,27,40)(19,28,22,31)(21,32,35,46)(24,37,36,30)(26,39,42,48)(29,43,34,45)(38,47,41,44);(2,7)(6,9)(8,14)(13,15)(17,21)(18,26)(19,29)(20,32)(22,34)(23,35)(24,38)(25,39)(27,42)(28,43)(30,44)(31,45)(33,46)(36,41)(37,47)(40,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(21,35)(26,42)(29,34)(32,46)(38,41)(39,48)(43,45)(44,47)Polycyclic presentation:
2^8|b^2=h,c^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cfh,d^b=dh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #508111, #508113, #6352014, … |
| maximal normal subgroup of | #508111, #508113, #6352014, … |
| maximal quotient of | #6374696, #6374699, #6375047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54006);