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 |
| unambiguous semidirect product | C22⋊#564 |
| 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 | |
933 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,13)(7,8,14,11)(9,12,17,19)(15,16,20,18)(21,22,27,25)(23,30,24,28)(26,33,34,29)(31,35,32,36);(7,14)(8,11)(9,17)(12,19)(21,23,27,24)(22,28,25,30)(26,31,34,32)(29,35,33,36);(1,2)(3,4)(5,7)(6,11)(8,13)(9,15)(10,14)(12,18)(16,19)(17,20)(21,24,27,23)(22,28,25,30)(26,32,34,31)(29,35,33,36);(5,8,10,11)(6,7,13,14)(9,16,17,18)(12,15,19,20)(21,25,27,22)(23,28,24,30)(26,33,34,29)(31,35,32,36);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,26,27,34)(22,29,25,33)(23,31,24,32)(28,35,30,36)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=f,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 | #97358, #246786, #246787, … |
| maximal normal subgroup of | #97358, #246786, #246787, … |
| maximal quotient of | #557046, #557366, #557572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39125);