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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2669 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,22)(21,24);(2,6)(3,8)(10,12)(11,15)(17,18)(19,23)(20,22)(21,24);(9,11,13,15)(10,14,12,16)(17,18,20,22)(19,21,24,23);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24);(1,4,5,7)(2,3,6,8)(9,12,13,10)(11,16,15,14)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=fgh,c^2=h,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×#44598, #7668930, #7669056, … |
| maximal normal subgroup of | C2×#44598, #7668930, #7669056, … |
| maximal quotient of | #535793, #535813, #536359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37586);