Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
927 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)(7,13)(11,16)(14,19)(18,20)(22,25)(23,29)(28,32)(30,35);(6,10)(7,13)(8,15)(18,20)(21,22,26,25)(23,27,29,31)(24,32,33,28)(30,36,35,34);(1,2)(3,4)(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,24,26,33)(22,28,25,32)(23,30,29,35)(27,34,31,36);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25,26,22)(23,31,29,27)(24,32,33,28)(30,36,35,34)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=h,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97326, #246788, #246795, … |
| maximal normal subgroup of | #97326, #246788, #246795, … |
| maximal quotient of | #557051, #557365, #557597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39217);