Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2407 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,8)(9,11)(10,12)(14,18)(15,20);(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,20)(17,18);(1,2)(3,4)(5,8,10,9)(6,11,12,7)(13,15,17,20)(14,19,18,16);(5,7)(6,8)(9,12)(10,11)(13,16,17,19)(14,20,18,15);(5,9,10,8)(6,11,12,7)Polycyclic presentation:
2^8|c^2=f,d^2=fh,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6577100, #6577101, #6577124, … |
| maximal normal subgroup of | #6577100, #6577101, #6577124, … |
| maximal quotient of | #517778, #517780, #517796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28712);