Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 131072 |
| 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 | #2541 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(6,9)(11,12)(14,17)(18,20);(1,2,4,3)(5,6,10,9)(7,12,8,11);(1,3,4,2)(5,7,10,8)(6,11,9,12)(13,14)(15,18)(16,17)(19,20);(5,8,10,7)(6,12,9,11);(5,9,10,6)(7,12,8,11)(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=h,e^2=fg,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #97646, #161586, #161870, … |
| maximal normal subgroup of | #97646, #161586, #161870, … |
| maximal quotient of | #516442, #516453, #516624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28186);