Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
544 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(5,6)(7,15,19,26)(8,20,23,29)(9,14)(10,13)(11,18,27,17)(12,22,28,21)(16,32)(24,30)(25,35)(31,33)(34,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,19)(12,25)(13,27)(14,26)(15,30)(18,24)(20,31)(21,33)(22,34)(23,35)(28,32)(29,36);(2,4)(5,8)(6,12)(7,16)(9,21)(10,23)(11,25)(13,28)(14,29)(15,31)(17,33)(18,34)(19,35)(20,30)(22,24)(26,36)(27,32);(6,13)(7,17)(9,24)(11,15)(12,28)(16,33)(18,19)(21,22)(25,31)(26,27)(32,36)(34,35)Polycyclic presentation:
2^8|a^2=efh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,f^c=fh,e^d=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #153132, #6667750, #6667751, … |
| maximal normal subgroup of | #153132, #6667750, #6667751, … |
| maximal quotient of | #343004, #343007, #343008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24060);