Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C28×D4 (order 2048) · Inn(G) = C22×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #449 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3,6,12)(2,7,8,15)(4,11,5,14)(9,16,13,10);(2,8)(7,15)(9,13)(10,16)(17,19,20,26,21,25,27,31)(18,22,23,29,24,28,30,32);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(19,25)(22,28)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eg,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18005, #6344906, #6344942, … |
| maximal normal subgroup of | C2×#18005, #6344906, #6344942, … |
| maximal quotient of | #182581, #183215, #183499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10993);