Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C42 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#97 |
| prime radical | C23 | #97 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,11,16,12,17,19,24,13,15,18,23,20,14,21,22);(1,2,3,5,4,6,7,8)(10,14,15,17)(11,18)(16,24,23,22)(19,21);(10,15)(14,17)(16,23)(22,24)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,b^a=bf,c^a=ch,e^a=eh,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #94121, C8×#347, #126535, … |
| maximal normal subgroup of | #94121, C8×#347, #126535, … |
| maximal quotient of | #122213, #122216, #122217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4392);