Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,16)(11,12,15,14)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(2,6)(4,7)(11,15)(12,14)(18,22)(23,25)(24,29)(30,31);(1,2)(3,7)(4,8)(5,6)(9,11,13,15)(10,14,16,12)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,3,5,8)(2,7,6,4)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,4,5,7)(2,8,6,3)(9,12,13,14)(10,15,16,11)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=g,c^2=fg,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#40158, #7637551, #7638128, … |
| maximal normal subgroup of | C2×#40158, #7637551, #7638128, … |
| maximal quotient of | #521994, #522530, #523091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33146);