Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 262144 |
| 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 | C2×#532 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,10,8)(7,12,9,11)(13,14)(15,17)(16,19)(18,20);(2,4)(5,7,10,9)(6,11,8,12)(14,15)(18,19);(13,14,17,15)(16,18,20,19);(1,2)(3,4)(5,8,10,6)(7,12,9,11)(13,15,17,14)(16,19,20,18);(5,9,10,7)(6,12,8,11)(13,16,17,20)(14,18,15,19)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=g,d^2=f,e^2=f,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 | #98606, #133343, #246442, … |
| maximal normal subgroup of | #98606, #133343, #246442, … |
| maximal quotient of | #531662, #532032, #532397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52072);