Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C2×#120 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,15)(8,21)(9,19)(10,16)(12,14)(13,28)(17,30)(20,25)(22,32)(23,29)(24,31)(26,27);(1,2,5,10)(3,8,12,22)(4,16,15,7)(6,9,18,24)(11,25,14,23)(13,29,27,20)(17,31,30,19)(21,26,32,28);(3,12)(4,17)(7,19)(8,22)(11,14)(13,26)(15,30)(16,31)(20,21)(23,25)(27,28)(29,32);(1,3)(2,8)(4,13)(5,12)(6,14)(7,20)(9,23)(10,22)(11,18)(15,27)(16,29)(17,28)(19,32)(21,31)(24,25)(26,30)Polycyclic presentation:
2^8|b^2=g,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18448, #6347063, #6347083, … |
| maximal normal subgroup of | C2×#18448, #6347063, #6347083, … |
| maximal quotient of | #168911, #168923, #168968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11436);