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 | |
766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,4)(5,7)(6,8)(9,10,13,12)(11,15,16,14)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30);(3,5)(4,7)(18,22)(20,29)(23,25)(26,32);(1,3,6,5)(2,7,8,4)(9,11,13,16)(10,14,12,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,4,6,7)(2,3,8,5)(9,10,13,12)(11,14,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,5,6,3)(2,7,8,4)(9,12,13,10)(11,15,16,14)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fgh,c^2=fg,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×#49481, #7704419, #7704979, … |
| maximal normal subgroup of | C2×#49481, #7704419, #7704979, … |
| maximal quotient of | #560766, #560980, #561889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42469);