Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
501 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,11,18,10,17)(7,15,9,28,21,30,20,13)(8,22,25,14,27,34,26,33)(19,31,35,32,23,29,24,36);(1,2,3,4)(5,7,10,20,11,21,12,9)(6,13,16,15,18,28,17,30)(8,23,25,24,27,19,26,35)(14,31,22,36,33,29,34,32);(2,4)(5,8)(6,14)(7,19)(9,24)(10,25)(11,27)(12,26)(13,29)(15,32)(16,22)(17,34)(18,33)(20,35)(21,23)(28,31)(30,36);(5,9,12,21,11,20,10,7)(6,15,17,13,18,30,16,28)(8,24,26,23,27,35,25,19)(14,32,34,29,33,36,22,31)Polycyclic presentation:
2^8|a^2=fh,b^2=egh,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139401, #139404, #143210, … |
| maximal normal subgroup of | #139401, #139404, #143210, … |
| maximal quotient of | #321513, #321517, #321522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21486);