Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 131072 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(11,15)(13,16)(17,18,22,23)(19,24,25,20)(21,29,30,26)(27,28,32,31);(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(18,23)(20,24)(26,29)(28,31);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28);(1,3,4,7)(2,6,5,8)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=fg,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133196, #133199, #251752, … |
| maximal normal subgroup of | #133196, #133199, #251752, … |
| maximal quotient of | #553769, #553793, #553991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30385);