Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
724 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 724 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(5,6,9,11,8,12,15,16)(7,10,14,18,13,19,17,20);(1,2)(3,4)(9,15)(11,16)(14,17)(18,20);(2,4)(5,6)(7,10)(8,12)(9,16)(11,15)(13,19)(14,20)(17,18);(5,7,8,13)(6,10,12,19)(9,17,15,14)(11,20,16,18)Polycyclic presentation:
2^8|a^2=f,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,e^c=eh,f^c=fh,f^d=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #501121, #501124, #501127, … |
| maximal normal subgroup of | #501121, #501124, #501127, … |
| maximal quotient of | #267471, #267479, #267487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17404);