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:
(2,5)(6,8)(10,14)(11,16)(17,18)(19,22)(20,24)(21,23);(1,2)(3,6)(4,5)(7,8)(10,14)(11,16)(18,19)(23,24);(9,10)(11,15)(12,16)(13,14)(17,19,22,18)(20,24,21,23);(1,3,4,7)(2,6,5,8)(9,11,13,16)(10,15,14,12)(17,20,22,21)(18,23,19,24);(1,2)(3,6)(4,5)(7,8)(9,12,13,15)(10,16,14,11)(17,21,22,20)(18,24,19,23)Polycyclic presentation:
2^8|c^2=gh,d^2=fh,e^2=g,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×#59968, #7666595, #7666666, … |
| maximal normal subgroup of | C2×#59968, #7666595, #7666666, … |
| maximal quotient of | #532153, #532160, #532202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52956);