Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1006 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(6,10)(7,8)(11,12);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11);(2,4)(5,6)(7,10)(8,12)(9,11);(5,6)(7,12)(8,10)(9,11)Polycyclic presentation:
2^8|e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,e^b=eh,f^c=fh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #273307, #273309, #273343, … |
| maximal normal subgroup of | #273307, #273309, #273343, … |
| maximal quotient of | #244655, #244656, #244657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16864);