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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(13,14)(15,16)(17,19)(18,20);(5,6)(7,11)(8,12)(9,10)(13,15,17,20)(14,18,19,16);(1,2,3,4)(6,10)(7,12)(14,19)(15,20);(5,7,9,12)(6,11,10,8)(13,14)(15,18)(16,20)(17,19);(5,8,9,11)(6,7,10,12)(13,16,17,18)(14,15,19,20)Polycyclic presentation:
2^8|b^2=f,c^2=fg,d^2=fh,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 | #98137, #164140, #164217, … |
| maximal normal subgroup of | #98137, #164140, #164217, … |
| maximal quotient of | #519068, #519078, #519090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29460);