Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18,22,23)(19,26,24,20)(21,25,29,30)(27,32,31,28);(1,2,4,6)(3,7,8,5)(9,11,13,10)(12,15,16,14)(18,23)(19,24)(25,30)(27,31);(1,2)(3,5)(4,6)(7,8)(9,12)(10,14)(11,15)(13,16)(17,19,22,24)(18,20,23,26)(21,27,29,31)(25,28,30,32);(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,30)(29,32);(1,3,4,8)(2,5,6,7)(17,21,22,29)(18,25,23,30)(19,27,24,31)(20,28,26,32)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=f,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 | #130656, #254340, #254363, … |
| maximal normal subgroup of | #130656, #254340, #254363, … |
| maximal quotient of | #521602, #522235, #522886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32139);