Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 131072 |
| 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 | #2408 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,18)(17,20);(1,2)(3,4)(6,10)(7,12)(13,14,16,18)(15,17,20,19);(5,7,9,12)(6,11,10,8)(13,14)(15,19)(16,18)(17,20);(5,8,9,11)(6,7,10,12)(13,15,16,20)(14,17,18,19)Polycyclic presentation:
2^8|c^2=fg,d^2=fgh,e^2=h,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 | #130857, #248501, #248502, … |
| maximal normal subgroup of | #130857, #248501, #248502, … |
| maximal quotient of | #519021, #519045, #519136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29516);