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)(10,13)(14,16)(17,18,22,25)(19,21,26,23)(20,24,30,32)(27,29,28,31);(1,3,6,5)(2,7,8,4)(19,26)(21,23)(27,28)(29,31);(1,3,6,5)(2,4,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32);(1,4,6,7)(2,3,8,5)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(1,5,6,3)(2,7,8,4)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,19,25,26)(20,29,30,31)(24,27,32,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fgh,d^2=f,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×#44646, #7669087, #7669105, … |
| maximal normal subgroup of | C2×#44646, #7669087, #7669105, … |
| maximal quotient of | #535846, #535898, #536467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37634);