Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C8×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4×C8 | C22 |
| commutator | C2 | C8×C16 |
| socle, prime radical | C23 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2)(3,4)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(2,4)(5,6,7,9,8,10,11,12)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,b^a=bc160 conjugacy classes — view the full character table.
| maximal subgroup of | #92821, #92822, #92869, … |
| maximal normal subgroup of | #92821, #92822, #92869, … |
| maximal quotient of | #92895, #92896, #92897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 124);