Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #72286249 (order 2048) · Inn(G) = C42 · Out(G) = C2×#525 |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C42 |
| commutator | C4 | C4×C16 |
| Frattini, cosocle | C22×C16 | C22 |
| socle | C22 | #346 |
| prime radical | C23 | M32 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
91 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(5,6,8,14,9,16,20,29,10,12,21,24,22,25,33,34,11,15,19,28,23,30,32,36,7,13,17,26,18,27,31,35);(2,4)(6,12,15,13)(8,19)(14,26,28,24)(16,25,30,27)(17,21)(20,32)(29,35,36,34)(31,33)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=f,f^2=g,g^2=h,b^a=bc,e^a=eh,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93888, #93891, #93921, … |
| maximal normal subgroup of | #93888, #93891, #93921, … |
| maximal quotient of | #93990, #93998, #94005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 454);