Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C22×C22⋊C4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #525 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,9,21)(4,15)(8,20,24,31)(10,22,11,16)(13,28)(17,19)(23,29,25,32)(26,30);(1,2,5,10)(3,8,14,25)(4,16,15,21)(6,11,18,9)(7,19,22,17)(12,23,27,24)(13,29,28,31)(20,30,32,26);(1,3,5,14)(2,8,10,25)(4,13,15,28)(6,12,18,27)(7,20,22,32)(9,24,11,23)(16,29,21,31)(17,26,19,30);(3,12)(8,23)(13,26)(14,27)(20,29)(24,25)(28,30)(31,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,e^2=h,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17609, #6343436, #6343443, … |
| maximal normal subgroup of | C2×#17609, #6343436, #6343443, … |
| maximal quotient of | #163729, #163731, #163736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10597);