Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2076 |
| prime radical | C24 | Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,16,23,30,32,6,9,18,24,5,10,17,25)(3,11,13,26,19,31,29,20,12,27,28,21,14,22,15,7);(7,20)(11,27)(21,26)(22,31);(1,3,5,19,6,12,16,14)(2,7,10,26,9,20,23,21)(4,15,17,13,18,29,30,28)(8,22,25,11,24,31,32,27)Polycyclic presentation:
2^8|a^2=e,c^2=dg,e^2=gh,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^c=dh,e^c=eg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93792, #93793, #151490, … |
| maximal normal subgroup of | #93792, #93793, #151490, … |
| maximal quotient of | #146023, #146033, #146057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6486);