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 | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(4,8)(9,10,12,14)(11,15,16,13)(18,21)(19,23);(1,2,5,6)(3,4,7,8)(9,11,12,16)(10,13,14,15)(17,18,20,21)(19,22,23,24);(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,21)(23,24);(1,3,5,7)(2,4,6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,23)(21,24);(1,4)(2,7)(3,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^8|a^2=h,b^2=f,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×#59268, #7665767, #7666084, … |
| maximal normal subgroup of | C2×#59268, #7665767, #7666084, … |
| maximal quotient of | #531632, #531644, #532104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52256);