Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(3,7)(11,15)(13,16)(17,18,22,23)(19,21,25,31)(20,27,30,24)(26,32,28,29);(1,2,5,4)(3,8,7,6)(9,10,12,14)(11,13,15,16)(18,23)(21,31)(24,27)(29,32);(1,3,5,7)(2,6,4,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,25)(18,21,23,31)(20,28,30,26)(24,32,27,29);(1,4,5,2)(3,8,7,6)(17,20,22,30)(18,24,23,27)(19,26,25,28)(21,29,31,32);(17,21)(18,25)(19,23)(20,29)(22,31)(24,28)(26,27)(30,32)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133205, #253310, #253312, … |
| maximal normal subgroup of | #133205, #253310, #253312, … |
| maximal quotient of | #544433, #545029, #545499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34381);