Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | |
774 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)(3,6)(4,7)(5,8)(10,13)(14,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31);(1,2,5,8)(3,6,4,7)(19,26)(21,29)(23,25)(24,28);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,2)(3,6)(4,7)(5,8)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(1,4,5,3)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fgh,d^2=h,e^2=fg,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#44865, #7669285, #7669293, … |
| maximal normal subgroup of | C2×#44865, #7669285, #7669293, … |
| maximal quotient of | #535791, #535808, #536358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37853);