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 | |
1315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,10)(8,11)(14,18)(15,16);(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,17,18)(15,19,16,20);(5,7,9,12)(6,11,10,8)(13,14)(15,19)(16,20)(17,18);(1,2)(3,4)(13,15,17,16)(14,19,18,20);(5,8,9,11)(6,12,10,7)(13,16,17,15)(14,20,18,19)Polycyclic presentation:
2^8|b^2=g,c^2=fg,d^2=g,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 | #6996823, #6996824, #6996839, … |
| maximal normal subgroup of | #6996823, #6996824, #6996839, … |
| maximal quotient of | #531700, #531742, #531769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52133);