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 | |
761 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(11,15)(14,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32);(1,3,6,4)(2,7,8,5)(9,10,13,12)(11,14,15,16)(18,22)(23,29)(24,27)(30,31);(2,8)(3,4)(9,11,13,15)(10,14,12,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(1,4,6,3)(2,7,8,5)(9,12,13,10)(11,16,15,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32);(1,5,6,7)(2,3,8,4)(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fg,d^2=g,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×#49091, #7704612, #7704966, … |
| maximal normal subgroup of | C2×#49091, #7704612, #7704966, … |
| maximal quotient of | #561226, #561383, #562297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42079);