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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,24)(21,29)(23,26)(25,31)(27,32)(28,30);(1,2)(3,7)(4,5)(6,8)(17,19,23,20)(18,24,26,22)(21,27,31,28)(25,32,29,30);(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20,23,19)(18,24,26,22)(21,27,31,28)(25,30,29,32);(1,2)(3,6)(4,5)(7,8)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,13,14,16)(17,22,23,24)(18,19,26,20)(21,30,31,32)(25,27,29,28)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=fg,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×#40964, #7639305, #7639586, … |
| maximal normal subgroup of | C2×#40964, #7639305, #7639586, … |
| maximal quotient of | #544739, #545228, #545761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33952);