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, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,3)(4,7,5,8)(11,15)(13,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32);(1,3,6,2)(4,7,5,8)(9,10,12,14)(11,13,15,16)(18,22)(23,29)(24,27)(30,31);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,13,14,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31);(1,5,6,4)(2,8,3,7)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=fgh,d^2=f,e^2=h,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 | #130671, #130674, #254208, … |
| maximal normal subgroup of | #130671, #130674, #254208, … |
| maximal quotient of | #521502, #522154, #522658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32746);