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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
801 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,7)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,21,23,24);(1,3,6,4)(2,7,5,8)(10,13)(14,15);(3,4)(7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24);(1,4,6,3)(2,7,5,8)(9,10)(11,14)(12,13)(15,16);(1,5,6,2)(3,7,4,8)(9,11)(10,14)(12,16)(13,15)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|a^2=fgh,b^2=h,d^2=h,e^2=fg,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×#51259, #7707224, #7707266, … |
| maximal normal subgroup of | C2×#51259, #7707224, #7707266, … |
| maximal quotient of | #560821, #561123, #561910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44247);