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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
831 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)(3,4)(5,7)(6,8)(10,14)(12,16)(17,18)(19,22)(20,24)(21,23);(1,3,5,8)(2,4,7,6)(9,10,13,14)(11,12,15,16)(18,19)(23,24);(1,2,5,7)(3,6,8,4)(9,11,13,15)(10,12,14,16)(17,19,22,18)(20,24,21,23);(1,3)(2,6)(4,7)(5,8)(17,20,22,21)(18,23,19,24);(1,4,5,6)(2,3,7,8)(9,12)(10,15)(11,14)(13,16)(17,21,22,20)(18,24,19,23)Polycyclic presentation:
2^8|b^2=fg,c^2=fh,d^2=gh,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×#60012, #7666595, #7666606, … |
| maximal normal subgroup of | C2×#60012, #7666595, #7666606, … |
| maximal quotient of | #532172, #532243, #532309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53000);