Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
577 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 577 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(6,11,14,13)(7,15,16,17);(1,2)(3,4)(5,6)(7,12)(8,13)(9,11)(10,14)(15,19)(16,20)(17,18);(2,4)(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,19)(14,20);(6,13)(8,9)(11,14)(12,19)(15,17)(18,20)Polycyclic presentation:
2^8|a^2=ef,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,f^c=fh,e^d=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125647, #125649, #125892, … |
| maximal normal subgroup of | #125647, #125649, #125892, … |
| maximal quotient of | #343003, #343006, #343010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24058);