Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#121 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
675 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 675 conjugacy classes of subgroups — too many to draw. Only its 167 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 — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,20)(13,25)(18,27)(19,29)(21,28)(23,35)(24,31)(26,30)(32,33)(34,36);(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,27)(14,17)(15,28)(16,29)(18,26)(22,32)(24,34)(25,33)(30,36)(31,35);(2,4)(5,8,9,24,11,26,10,25)(6,13,15,30,16,31,14,22)(7,18,20,34,21,23,19,33)(12,27,17,32,29,35,28,36);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,24,26,25)(12,17,29,28)(13,22,31,30)(18,33,23,34)(27,32,35,36)Polycyclic presentation:
2^8|c^2=e,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #359166, #359167, #376299, … |
| maximal normal subgroup of | #359166, #359167, #376299, … |
| maximal quotient of | #238983, #238987, #238988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16518);