Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C2×C4×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C22 |
| commutator | C2 | C42×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C23 | C2×C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(5,6,7,9,8,10,11,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(1,2,3,4)(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,c^b=cg160 conjugacy classes — view the full character table.
| maximal subgroup of | #93259, #93265, #93267, … |
| maximal normal subgroup of | #93259, #93265, #93267, … |
| maximal quotient of | C8×#322, #94852, #94854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 829);