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 | #2619 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,8,4)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32);(1,3,5,8)(2,6,7,4)(9,10)(11,15)(12,16)(13,14)(18,23)(20,24)(26,29)(28,31);(2,7)(3,8)(10,14)(11,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(1,4,5,6)(2,3,7,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,d^2=f,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×#48417, #7703942, #7704111, … |
| maximal normal subgroup of | C2×#48417, #7703942, #7704111, … |
| maximal quotient of | #560713, #560941, #561829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41405);