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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
707 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)(3,4)(5,7)(6,8)(11,15)(14,16)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30);(1,3,6,5)(2,7,8,4)(9,10,13,12)(11,14,15,16)(18,22)(19,25)(20,29)(30,32);(1,3,6,5)(2,4,8,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,4,6,7)(2,3,8,5)(9,12,13,10)(11,16,15,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31);(1,5,6,3)(2,7,8,4)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=fh,d^2=fg,e^2=f,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×#44528, #7669087, #7669101, … |
| maximal normal subgroup of | C2×#44528, #7669087, #7669101, … |
| maximal quotient of | #535634, #535718, #536319, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37516);