Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
826 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 826 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,14)(8,18)(9,12)(10,13)(11,17)(15,16)(19,20);(1,2,4,3)(5,6)(7,15)(8,12)(9,18)(10,13)(11,17)(14,16)(19,20);(5,7,8,17,10,16,9,19)(6,11,12,14,13,20,18,15);(1,3,4,2)(7,16)(11,20)(14,15)(17,19)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #250327, #250329, #250335, … |
| maximal normal subgroup of | #250327, #250329, #250335, … |
| maximal quotient of | #297407, #297410, #297415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18737);