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 | |
754 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)(3,8)(4,5)(6,7)(10,14)(15,16)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30);(1,3,6,4)(2,5,7,8)(19,24)(20,23)(21,29)(25,32);(2,7)(3,4)(9,10,13,14)(11,15,12,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,30)(28,32)(29,31);(1,4,6,3)(2,5,7,8)(9,11,13,12)(10,15,14,16)(17,20,22,23)(18,24,26,19)(21,28,29,30)(25,31,32,27);(1,5)(2,3)(4,7)(6,8)(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=gh,d^2=g,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×#50581, #7706031, #7706232, … |
| maximal normal subgroup of | C2×#50581, #7706031, #7706232, … |
| maximal quotient of | #560784, #561089, #561872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43569);