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 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(9,10,13,16)(11,14,12,15)(17,18)(19,24)(20,25)(21,28)(22,23)(26,31)(27,30)(29,32);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,14,16,15)(18,23)(19,27)(20,29)(21,31);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,19,22,27)(18,24,23,30)(20,26,29,28)(21,25,31,32);(1,3,5,8)(2,6,4,7)(17,20,22,29)(18,25,23,32)(19,28,27,26)(21,30,31,24);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,16,14)(17,21)(18,26)(19,25)(20,30)(22,31)(23,28)(24,29)(27,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=f,d^2=f,e^2=g,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×#46992, #7700720, #7700752, … |
| maximal normal subgroup of | C2×#46992, #7700720, #7700752, … |
| maximal quotient of | #576842, #576890, #577092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39980);