Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C2×#127 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,17,19,16,18,15,20);(5,7,10,6)(8,11,9,12)(13,14,15,20,16,18,17,19);(1,2,3,4)(5,8,10,9)(6,11,7,12)(14,18)(19,20);(5,9,10,8)(6,11,7,12)(13,15,16,17)(14,19,18,20)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #100651, #100655, #100659, … |
| maximal normal subgroup of | #100651, #100655, #100659, … |
| maximal quotient of | #219444, #219445, #219446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15254);