Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(10,13)(14,15)(20,28)(23,29)(26,32)(30,31);(9,10)(11,15)(12,13)(14,16)(17,18,21,24)(19,22,27,25)(20,29,28,23)(26,31,32,30);(1,3,5,8)(2,6,4,7)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,4,5,2)(3,7,8,6)(9,11,12,16)(10,14,13,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,32)(29,31);(17,20,21,28)(18,23,24,29)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=fg,e^2=h,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 | #130603, #131024, #253946, … |
| maximal normal subgroup of | #130603, #131024, #253946, … |
| maximal quotient of | #535406, #535512, #536127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38294);