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 | |
3141 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(18,23)(21,33)(24,27)(25,34)(29,43)(32,44)(36,39)(42,48);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(20,31)(25,34)(28,41)(32,44)(36,39)(38,45)(42,48)(46,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,22,23)(19,27,30,24)(20,25,31,34)(21,26,33,35)(28,39,41,36)(29,40,43,37)(32,38,44,45)(42,47,48,46);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,30)(18,24,23,27)(20,28,31,41)(21,29,33,43)(25,36,34,39)(26,37,35,40)(32,42,44,48)(38,46,45,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,28)(21,32)(22,31)(23,34)(24,36)(26,38)(27,39)(29,42)(30,41)(33,44)(35,45)(37,46)(40,47)(43,48);(17,21)(18,26)(19,29)(20,32)(22,33)(23,35)(24,37)(25,38)(27,40)(28,42)(30,43)(31,44)(34,45)(36,46)(39,47)(41,48)Polycyclic presentation:
2^8|b^2=g,c^2=h,d^2=h,e^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 | #346489, #346499, #346511, … |
| maximal normal subgroup of | #346489, #346499, #346511, … |
| maximal quotient of | #7626363, #7626364, #7626400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55861);