Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,5)(3,8,4,7)(17,18)(19,23)(20,22)(21,24);(2,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,21,24,23);(1,3,6,4)(2,7,5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24);(1,4,6,3)(2,8,5,7)(17,19,20,24)(18,21,22,23);(1,5,6,2)(3,8,4,7)(9,11,12,16)(10,13,14,15)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98344, #266294, #266298, … |
| maximal normal subgroup of | #98344, #266294, #266298, … |
| maximal quotient of | #521622, #522263, #522895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32168);