Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(13,14,15,18,16,17,19,20);(1,2,3,4)(6,9)(10,12);(5,7,8,11)(6,10,9,12);(14,17)(18,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=e,f^2=h,b^a=be,d^a=dh112 conjugacy classes — view the full character table.
| maximal subgroup of | #94864, #94865, #94872, … |
| maximal normal subgroup of | #94864, #94865, #94872, … |
| maximal quotient of | #181815, #181847, #181850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13352);