Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(10,13)(14,15)(17,18)(19,22)(20,23)(21,26)(24,29)(25,27)(28,31)(30,32);(2,4)(6,7)(9,10,12,13)(11,15,16,14)(18,19)(20,27)(21,29)(30,31);(1,3)(2,6)(4,7)(5,8)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,4,5,2)(3,7,8,6)(9,11,12,16)(10,14,13,15)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=h,d^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130580, #254385, #254432, … |
| maximal normal subgroup of | #130580, #254385, #254432, … |
| maximal quotient of | #535340, #535444, #535961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37450);