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×#121 | C22 |
| 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) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,13)(11,16,15,12)(17,18,22,20)(19,24,21,23);(1,2,4,6)(3,5,8,7)(11,15)(12,16)(17,19,22,21)(18,23,20,24);(9,11)(10,12)(13,16)(14,15)(17,20,22,18)(19,24,21,23);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,15,13,11);(1,3,4,8)(2,5,6,7)(9,13,14,10)(11,16,15,12)(17,21,22,19)(18,24,20,23)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fgh,d^2=g,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×#59797, #7666285, #7666425, … |
| maximal normal subgroup of | C2×#59797, #7666285, #7666425, … |
| maximal quotient of | #532079, #532296, #532331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52785);