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 | |
1270 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)(5,6,10,12)(7,9,8,11);(5,7,10,8)(6,11,12,9)(13,14)(15,19)(16,20)(17,18);(1,2)(3,4)(6,12)(7,8)(14,18)(15,20);(5,8,10,7)(6,11,12,9)(13,15)(14,16)(17,20)(18,19);(5,9,10,11)(6,7,12,8)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=f,e^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×#47845, #7703942, #7704031, … |
| maximal normal subgroup of | C2×#47845, #7703942, #7704031, … |
| maximal quotient of | #560321, #560340, #561418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40833);