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 | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,12)(14,15)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32);(1,2,5,4)(3,6,7,8)(9,10,13,12)(11,15,16,14)(18,21)(23,25)(24,28)(31,32);(1,3)(2,6)(4,8)(5,7)(9,11,13,16)(10,14,12,15)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,32,28,31);(9,12,13,10)(11,15,16,14)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32);(1,4,5,2)(3,8,7,6)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^2=f,d^2=f,e^2=g,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×#41659, #7639886, #7640239, … |
| maximal normal subgroup of | C2×#41659, #7639886, #7640239, … |
| maximal quotient of | #544791, #545243, #545868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34647);