Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 524288 |
| 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 | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1247 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(7,11)(10,12)(13,14)(15,16)(17,18)(19,20);(1,2)(3,4)(6,9)(10,12)(14,18)(15,20);(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,19,18,16);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18);(5,7,8,11)(6,10,9,12)(13,16,17,19)(14,20,18,15)Polycyclic presentation:
2^8|a^2=fg,c^2=g,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #111673, #113503, #134198, … |
| maximal normal subgroup of | #111673, #113503, #134198, … |
| maximal quotient of | #516433, #516461, #516589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28268);