Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,8,10)(7,11,12,9)(14,17)(15,18)(19,20);(5,7,8,12)(6,9,10,11);(13,14,18,20,16,19,15,17);(1,2,3,4)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=fh,d^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #102226, #102227, #102228, … |
| maximal normal subgroup of | #102226, #102227, #102228, … |
| maximal quotient of | #353041, #353043, #353045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26300);