Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| 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 | |
3140 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)(3,8,10,5)(4,12,14,9)(11,13,16,15)(17,18,23,28)(19,29,33,24)(20,21,26,25)(22,38,39,27)(30,31,34,36)(32,40,46,43)(35,41,42,37)(44,48,47,45);(2,7)(3,10)(9,12)(11,16)(17,19,23,33)(18,24,28,29)(20,30,26,34)(21,36,25,31)(22,32,39,46)(27,40,38,43)(35,44,42,47)(37,48,41,45);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19)(18,24)(20,34)(21,31)(22,32)(23,33)(25,36)(26,30)(27,40)(28,29)(35,47)(37,45)(38,43)(39,46)(41,48)(42,44);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,20,23,26)(18,25,28,21)(19,30,33,34)(22,35,39,42)(24,31,29,36)(27,41,38,37)(32,44,46,47)(40,45,43,48);(17,21,23,25)(18,26,28,20)(19,31,33,36)(22,37,39,41)(24,30,29,34)(27,42,38,35)(32,45,46,48)(40,44,43,47);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22,23,39)(18,27,28,38)(19,32,33,46)(20,35,26,42)(21,37,25,41)(24,40,29,43)(30,44,34,47)(31,45,36,48)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh70 conjugacy classes — view the full character table.
| maximal subgroup of | #511850, #511851, #10580604, … |
| maximal normal subgroup of | #511850, #511851, #10580604, … |
| maximal quotient of | #7626377, #7626385, #7626420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55879);