Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,7,8,5)(10,13)(14,16)(18,22)(20,28)(23,25)(26,32);(1,3)(2,7)(4,5)(6,8)(17,18,21,22)(19,25,27,23)(20,29,28,24)(26,30,32,31);(2,4)(3,8)(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,4,6,2)(3,5,8,7)(9,11,12,15)(10,14,13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,5)(2,3)(4,8)(6,7)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47498, #7700916, #7700968, … |
| maximal normal subgroup of | C2×#47498, #7700916, #7700968, … |
| maximal quotient of | #576797, #576865, #577119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40486);