Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C16 | C22 |
| commutator | C2 | C2×C82 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×C4×C8 |
| prime radical | C23 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,12,16,13,17,22,28,14,18,23,29,24,30,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(1,2,3,5,4,6,7,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,26)(17,27)(18,25)(22,33)(23,31)(24,32)(28,37)(29,35)(30,36)(34,39)(38,40);(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,c^b=ch160 conjugacy classes — view the full character table.
| maximal subgroup of | #93993, #93994, #94017, … |
| maximal normal subgroup of | #93993, #93994, #94017, … |
| maximal quotient of | C16×M32, #122206, #122207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4387);