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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,9)(7,8)(14,18)(15,20);(2,4)(5,6,10,9)(7,12,8,11)(14,18)(15,20);(5,7,10,8)(6,11,9,12)(13,14)(15,19)(16,20)(17,18);(1,2)(3,4)(5,8,10,7)(6,12,9,11)(13,15,17,20)(14,19,18,16);(5,9,10,6)(7,12,8,11)(13,16,17,19)(14,20,18,15)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=g,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 | #133328, #499847, #500173, … |
| maximal normal subgroup of | #133328, #499847, #500173, … |
| maximal quotient of | #531721, #532333, #532417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52172);