Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
853 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 853 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(5,6,10,14)(7,11,16,26)(8,18,19,30)(9,21,24,13)(12,23,28,20)(15,31,32,35)(17,29,22,27)(25,34,36,33);(1,2)(3,4)(5,7,10,16)(6,11,14,26)(8,15,19,32)(9,22,24,17)(12,25,28,36)(13,29,21,27)(18,35,30,31)(20,34,23,33);(2,4)(5,8)(6,12)(7,15)(9,23)(10,19)(11,25)(13,30)(14,28)(16,32)(17,34)(18,21)(20,24)(22,33)(26,36)(27,35)(29,31);(7,16)(9,24)(11,26)(13,21)(15,32)(18,30)(20,23)(25,36)Polycyclic presentation:
2^8|a^2=h,b^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139735, #139753, #144392, … |
| maximal normal subgroup of | #139735, #139753, #144392, … |
| maximal quotient of | #245170, #245175, #245182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16902);