Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
727 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(14,16)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28);(1,2,5,6)(3,4,7,8)(18,23)(20,27)(25,29)(28,32);(2,6)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32);(1,3)(2,4)(5,7)(6,8)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32);(1,4)(2,3)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130185, #485778, #485850, … |
| maximal normal subgroup of | #130185, #485778, #485850, … |
| maximal quotient of | #576921, #577164, #577315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40128);