Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62972 |
| 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 | |
3161 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)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,23)(21,25)(22,32)(24,30)(26,36)(28,39)(29,38)(31,33)(34,37)(35,44)(40,45)(41,43)(42,47)(46,48);(3,10)(4,12)(7,9)(8,13)(18,20)(21,33)(22,36)(24,27)(28,34)(29,43)(40,44)(47,48);(2,6)(7,9)(8,13)(14,15)(17,19,23,30)(18,24,20,27)(21,34,33,28)(22,29,36,43)(25,39,31,37)(26,38,32,41)(35,46,45,42)(40,48,44,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,23,18)(19,24,30,27)(21,31,33,25)(22,32,36,26)(28,37,34,39)(29,38,43,41)(35,44,45,40)(42,47,46,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21)(18,25)(19,28)(20,31)(22,35)(23,33)(24,37)(26,40)(27,39)(29,42)(30,34)(32,44)(36,45)(38,47)(41,48)(43,46);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,22)(18,26)(19,29)(20,32)(21,35)(23,36)(24,38)(25,40)(27,41)(28,42)(30,43)(31,44)(33,45)(34,46)(37,47)(39,48)Polycyclic presentation:
2^8|c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh67 conjugacy classes — view the full character table.
| maximal subgroup of | #347035, #347425, #347426, … |
| maximal normal subgroup of | #347035, #347425, #347426, … |
| maximal quotient of | #7627620, #7627622, #7627627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55965);