Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
841 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)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,31)(21,26)(22,35)(23,27)(25,32)(28,43)(29,38)(30,39)(33,41)(34,40)(36,42)(37,46)(44,47)(45,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,23,30)(18,24,27,39)(20,28,33,45)(21,34,22,29)(25,37,42,47)(26,40,35,38)(31,43,41,48)(32,46,36,44);(3,10)(6,9)(11,16)(13,15)(17,20,23,33)(18,25,27,42)(19,28,30,45)(21,32,22,36)(24,37,39,47)(26,41,35,31)(29,44,34,46)(38,43,40,48);(2,7)(6,9)(8,14)(13,15)(17,21,23,22)(18,26,27,35)(19,29,30,34)(20,32,33,36)(24,38,39,40)(25,41,42,31)(28,44,45,46)(37,43,47,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,30)(24,39)(28,45)(29,34)(37,47)(38,40)(43,48)(44,46)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,d^b=dh,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#60999, #10575845, #10575887, … |
| maximal normal subgroup of | C2×#60999, #10575845, #10575887, … |
| maximal quotient of | #6374685, #6374706, #6375089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53987);