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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
817 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)(3,7)(4,8)(5,6)(10,13)(14,16)(17,18,21,22)(19,25,26,23)(20,24,28,30)(27,32,29,31);(2,6)(7,8)(18,22)(19,26)(20,28)(31,32);(1,3,5,4)(2,7,6,8)(9,10,12,13)(11,14,15,16)(17,19,21,26)(18,23,22,25)(20,29,28,27)(24,32,30,31);(1,2)(3,7)(4,8)(5,6)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,32)(26,29);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)Polycyclic presentation:
2^8|a^2=h,c^2=fgh,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×#44640, #7668944, #7669036, … |
| maximal normal subgroup of | C2×#44640, #7668944, #7669036, … |
| maximal quotient of | #535786, #535789, #536443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37628);