Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(11,15)(13,16)(17,18,22,23)(19,26,24,20)(21,25,29,30)(27,32,31,28);(1,2)(3,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(18,23)(19,24)(25,30)(27,31);(2,6)(3,7)(9,11,12,15)(10,13,14,16)(17,19,22,24)(18,20,23,26)(21,27,29,31)(25,28,30,32);(1,3,5,7)(2,4,6,8)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,30)(29,32);(1,4)(2,7)(3,6)(5,8)(17,21,22,29)(18,25,23,30)(19,27,24,31)(20,28,26,32)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#42020, #7640503, #7640512, … |
| maximal normal subgroup of | C2×#42020, #7640503, #7640512, … |
| maximal quotient of | #544534, #545118, #545398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35008);