Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,14)(9,12)(10,11)(15,19)(16,20)(17,18)(24,32)(27,34)(30,36)(33,35);(6,11)(7,14)(12,18)(15,20)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(1,2)(3,4)(5,7)(6,8)(9,15)(10,14)(11,13)(12,16)(17,20)(18,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,8)(6,7)(9,16)(10,13)(11,14)(12,15)(17,19)(18,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,9,10,17)(6,12,11,18)(7,15,14,20)(8,16,13,19)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)Polycyclic presentation:
2^8|a^2=fg,b^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #112010, #112326, #113514, … |
| maximal normal subgroup of | #112010, #112326, #113514, … |
| maximal quotient of | #556344, #556345, #556368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30539);