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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,7)(5,8)(9,10,13,16)(11,12,15,14)(17,18,21,23)(19,20,22,24);(1,2)(3,5)(4,8)(6,7)(9,10,13,16)(11,12,15,14)(17,19)(18,20)(21,22)(23,24);(1,3,6,4)(2,5,7,8)(9,11,13,15)(10,14,16,12);(1,4,6,3)(2,8,7,5)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24);(1,5,6,8)(2,3,7,4)(9,12,13,14)(10,15,16,11)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fg,d^2=f,e^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×#59358, #7666705, #7666823, … |
| maximal normal subgroup of | C2×#59358, #7666705, #7666823, … |
| maximal quotient of | #531748, #531759, #532266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52346);