Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,7,4,5)(10,13)(14,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,30,32,31);(3,4)(5,7)(18,22)(20,29)(23,25)(26,32);(1,3,6,4)(2,5,8,7)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,4,6,3)(2,7,8,5)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,5,6,7)(2,3,8,4)(9,11,12,15)(10,14,13,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=h,c^2=fg,d^2=fh,e^2=g,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 | C2×#46201, #7702765, #7702847, … |
| maximal normal subgroup of | C2×#46201, #7702765, #7702847, … |
| maximal quotient of | #557211, #557223, #557540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39189);