Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | #2635 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
822 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)(3,5)(4,7)(6,8)(10,13)(11,12)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30);(2,3)(4,8)(9,10,14,13)(11,16,12,15)(18,19)(20,27)(21,29)(30,31);(2,3)(6,7)(9,11,14,12)(10,15,13,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,16,13,15)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,16,12,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=fg,b^2=fgh,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132766, #487517, #488229, … |
| maximal normal subgroup of | #132766, #487517, #488229, … |
| maximal quotient of | #585530, #587012, #587653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50370);