Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #380 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,15)(8,12,20,26)(9,21)(10,14)(11,17)(13,22)(16,27,30,36)(18,28)(19,32,33,25)(23,34,35,29)(24,31);(1,2)(3,4)(5,7,9,17,10,18,22,31)(6,11,13,15,14,24,21,28)(8,16,19,29,20,30,33,34)(12,23,25,27,26,35,32,36);(2,4)(5,8,9,19,10,20,22,33)(6,12,13,25,14,26,21,32)(7,16,17,29,18,30,31,34)(11,23,15,27,24,35,28,36)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^c=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131411, #131412, #144748, … |
| maximal normal subgroup of | #131411, #131412, #144748, … |
| maximal quotient of | #120172, #120190, #120191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4229);