Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | #2635 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(11,15)(14,16)(18,22)(20,28)(23,25)(26,32);(1,3,5,4)(2,6,8,7)(9,10,13,12)(11,14,15,16)(18,22)(23,25)(24,29)(30,31);(1,2,5,8)(3,6,4,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,2)(3,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,4,5,3)(2,7,8,6)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=f,b^2=fgh,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131214, #484884, #485447, … |
| maximal normal subgroup of | #131214, #484884, #485447, … |
| maximal quotient of | #556457, #556459, #556464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30566);