Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1425 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)(7,9)(8,11)(10,12)(13,14)(15,16)(17,19)(18,20);(2,4)(5,7,10,11)(6,8,12,9)(13,15,18,17)(14,16,20,19);(1,2)(3,4)(5,8,10,9)(6,11,12,7)(13,16,18,19)(14,17,20,15);(2,4)(5,6)(7,8)(9,11)(10,12)(13,17,18,15)(14,19,20,16);(1,2)(3,4)(5,9,10,8)(6,7,12,11)(13,14)(15,19)(16,17)(18,20)Polycyclic presentation:
2^8|b^2=fg,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #130825, #220001, #220002, … |
| maximal normal subgroup of | #130825, #220001, #220002, … |
| maximal quotient of | #519244, #519247, #519421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29137);