Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 8192 |
| 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 | C22×C4⋊C4 |
| prime radical | C2×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18,21,24,22,25,20,30)(19,26,28,31,29,32,27,23);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19,21,28,22,29,20,27)(18,23,30,32,25,31,24,26);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16);(3,10)(8,15)(11,14)(12,16)Polycyclic presentation:
2^8|a^2=f,b^2=eg,e^2=g,f^2=h,b^a=be,e^a=eg,d^c=dgh70 conjugacy classes — view the full character table.
| maximal subgroup of | #136454, #136456, #137041, … |
| maximal normal subgroup of | #136454, #136456, #137041, … |
| maximal quotient of | #180937, #183420, #183422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10824);