Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,10,6,8)(5,12,14,9)(11,15,13,16)(18,23)(24,34)(25,35)(26,36)(37,45)(38,46)(39,40)(47,48);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(18,23)(19,27)(25,35)(26,36)(28,41)(29,42)(39,40)(43,44);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,34)(28,37)(29,38)(30,39)(31,35)(32,40)(33,36)(41,45)(42,46)(43,47)(44,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,34)(25,37)(26,38)(30,43)(31,41)(32,44)(33,42)(35,45)(36,46)(39,47)(40,48);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,20,22,31)(18,25,23,35)(19,28,27,41)(21,32,33,30)(24,37,34,45)(26,40,36,39)(29,44,42,43)(38,48,46,47);(17,21,22,33)(18,26,23,36)(19,29,27,42)(20,30,31,32)(24,38,34,46)(25,39,35,40)(28,43,41,44)(37,47,45,48)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh70 conjugacy classes — view the full character table.
| maximal subgroup of | #347453, #347473, #347501, … |
| maximal normal subgroup of | #347453, #347473, #347501, … |
| maximal quotient of | #7627416, #7627417, #7627450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56050);