Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | Aut(G) order 32768 |
| 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 | |
949 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 949 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:
(7,13)(8,15)(10,18)(12,16)(14,17)(19,20);(1,2)(3,4)(5,6,9,11)(7,10,14,19)(8,16,15,12)(13,20,17,18);(2,4)(5,7)(6,10)(8,17)(9,14)(11,19)(12,20)(13,15)(16,18);(6,11)(8,15)(10,19)(13,17)Polycyclic presentation:
2^8|b^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139734, #139754, #144390, … |
| maximal normal subgroup of | #139734, #139754, #144390, … |
| maximal quotient of | #245169, #245174, #245181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16901);