Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | |
1211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(5,6,9,11)(7,12,8,10);(5,7,9,8)(6,10,11,12)(14,17)(15,19);(1,2)(3,4)(7,8)(10,12)(13,14,16,17)(15,18,19,20);(5,8,9,7)(6,10,11,12)(13,14)(15,18)(16,17)(19,20);(5,6)(7,12)(8,10)(9,11)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=fh,d^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#49572, #7705301, #7705663, … |
| maximal normal subgroup of | C2×#49572, #7705301, #7705663, … |
| maximal quotient of | #560505, #560557, #561596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42560);