Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C4⋊C4 |
| prime radical | C23 | #117 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(10,14,16,19)(15,21,22,23);(1,2,3,5,4,6,7,8)(9,10)(11,15)(12,19)(13,16)(14,20)(17,23)(18,22)(21,24);(9,11,12,17,13,18,20,24)(10,15,19,23,16,22,14,21)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=g,f^2=h,b^a=bd,d^b=dg,e^b=eg112 conjugacy classes — view the full character table.
| maximal subgroup of | #93264, #93265, #93266, … |
| maximal normal subgroup of | #93264, #93265, #93266, … |
| maximal quotient of | C8×#334, C8×#335, #96888, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1254);