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 | #392 | 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:
(1,2,5,3)(4,7,8,6)(10,14)(12,16)(17,18,21,24)(19,22,20,23);(2,3)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19,21,20)(18,22,24,23);(10,14)(11,15)(17,18)(19,22)(20,23)(21,24);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,12,14,16)(17,19,21,20)(18,22,24,23);(1,4)(2,6)(3,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,d^2=fg,e^2=g,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×#49429, #7704363, #7704981, … |
| maximal normal subgroup of | C2×#49429, #7704363, #7704981, … |
| maximal quotient of | #560863, #560979, #561971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42417);