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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
777 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(11,15)(13,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,32)(26,30)(28,31);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(18,22)(23,30)(24,27)(31,32);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31);(1,4,5,8)(2,6,3,7)(17,20,21,29)(18,24,22,27)(19,25,26,28)(23,31,30,32);(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)Polycyclic presentation:
2^8|a^2=h,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 | #133192, #253286, #253288, … |
| maximal normal subgroup of | #133192, #253286, #253288, … |
| maximal quotient of | #544445, #545019, #545506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34380);