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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,4,7,8)(10,13)(14,16)(17,18,22,19)(20,24,21,23);(2,6)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,23,21,24);(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,23,19,24);(1,3)(2,4)(5,7)(6,8)(17,21,22,20)(18,24,19,23);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fh,d^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×#59661, #7665934, #7666085, … |
| maximal normal subgroup of | C2×#59661, #7665934, #7666085, … |
| maximal quotient of | #531888, #531927, #532056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52649);