Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,7)(6,8)(9,10,13,12)(11,15,16,14);(1,3,5,8)(2,4,7,6)(9,11)(10,14)(12,15)(13,16)(17,18,20,22)(19,23,24,21);(1,3)(2,6)(4,7)(5,8)(9,10,13,12)(11,14,16,15);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,19,20,24)(18,21,22,23);(1,4,5,6)(2,3,7,8)(9,11)(10,14)(12,15)(13,16)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,d^2=fgh,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 | #7091177, #7091191, #7091242, … |
| maximal normal subgroup of | #7091177, #7091191, #7091242, … |
| maximal quotient of | #531915, #531916, #532216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52922);