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 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,10,8)(7,12,9,11)(13,14,16,18)(15,19,20,17);(2,4)(5,7,10,9)(6,11,8,12)(15,20)(17,19);(5,8,10,6)(7,12,9,11)(13,15)(14,17)(16,20)(18,19);(1,2)(3,4)(13,14)(15,17)(16,18)(19,20);(5,9,10,7)(6,12,8,11)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^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 | #6994430, #6994434, #6994494, … |
| maximal normal subgroup of | #6994430, #6994434, #6994494, … |
| maximal quotient of | #531615, #531628, #531733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52008);