Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #2678 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,8,5,7)(10,14)(15,16)(17,18)(19,23)(20,22)(21,24);(3,5)(7,8)(17,18,20,22)(19,21,24,23);(1,3,6,5)(2,7,4,8)(9,10,13,14)(11,15,12,16)(17,18)(19,21)(20,22)(23,24);(1,4,6,2)(3,7,5,8)(9,11,13,12)(10,15,14,16)(17,19)(18,21)(20,24)(22,23);(1,5,6,3)(2,7,4,8)(9,12,13,11)(10,16,14,15)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=g,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41859, #7639689, #7640245, … |
| maximal normal subgroup of | C2×#41859, #7639689, #7640245, … |
| maximal quotient of | #544969, #545349, #545919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34847);