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 | |
3143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,9,12)(5,8,13,14)(10,11,15,16)(17,18)(19,28)(20,21)(22,27)(23,24)(25,32)(26,33)(29,39)(30,42)(31,43)(34,35)(36,41)(37,38)(40,47)(44,46)(45,48);(2,7)(3,9)(8,14)(10,15)(18,24)(20,33)(25,28)(27,38)(29,42)(34,41)(40,43)(44,48);(1,3)(2,4)(5,10)(6,9)(7,12)(8,11)(13,15)(14,16)(17,19,23,32)(18,25,24,28)(20,29,33,42)(21,30,26,39)(22,31,37,47)(27,40,38,43)(34,44,41,48)(35,45,36,46);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,20)(18,26)(19,29)(21,24)(22,34)(23,33)(25,39)(27,36)(28,30)(31,44)(32,42)(35,38)(37,41)(40,46)(43,45)(47,48);(1,5,6,13)(2,8,7,14)(3,10,9,15)(4,11,12,16)(17,21,23,26)(18,20,24,33)(19,30,32,39)(22,36,37,35)(25,29,28,42)(27,41,38,34)(31,46,47,45)(40,48,43,44);(17,22)(18,27)(19,31)(20,34)(21,35)(23,37)(24,38)(25,40)(26,36)(28,43)(29,44)(30,45)(32,47)(33,41)(39,46)(42,48)Polycyclic presentation:
2^8|a^2=gh,c^2=h,e^2=g,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 | #511858, #511860, #6351949, … |
| maximal normal subgroup of | #511858, #511860, #6351949, … |
| maximal quotient of | #7627418, #7627419, #7627495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56052);