Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(10,14)(15,16)(17,18,20,21)(19,23,24,22);(2,3)(6,7)(9,10)(11,15)(12,16)(13,14)(18,21)(22,23);(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,21)(23,24);(1,3,5,2)(4,7,8,6)(9,12,13,11)(10,16,14,15)(17,19)(18,22)(20,24)(21,23);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^8|a^2=gh,c^2=fgh,d^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#59707, #7666168, #7666417, … |
| maximal normal subgroup of | C2×#59707, #7666168, #7666417, … |
| maximal quotient of | #532041, #532130, #532203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52695);