Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
423 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 129 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:
(5,6,12,18)(7,15,21,29)(8,22,25,32)(9,26,20,13)(10,17,11,16)(14,27,31,24)(19,33,35,28)(23,36,34,30);(1,2,3,4)(5,7,10,20,12,21,11,9)(6,13,16,15,18,26,17,29)(8,19,24,34,25,35,27,23)(14,28,22,30,31,33,32,36);(2,4)(5,8)(6,14)(7,19)(9,23)(10,24)(11,27)(12,25)(13,28)(15,30)(16,22)(17,32)(18,31)(20,34)(21,35)(26,33)(29,36);(5,9,11,21,12,20,10,7)(6,15,17,13,18,29,16,26)(8,23,27,35,25,34,24,19)(14,30,32,28,31,36,22,33)Polycyclic presentation:
2^8|a^2=h,b^2=eg,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #143702, #143706, #143708, … |
| maximal normal subgroup of | #143702, #143706, #143708, … |
| maximal quotient of | #317948, #317950, #317953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21229);