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 | #3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
847 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 847 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)(7,15,17,29)(8,12,19,27)(9,21,22,13)(11,23,26,18)(16,32,33,31)(20,30,24,28)(25,35,36,34);(1,2)(3,4)(8,19)(9,22)(12,27)(13,21)(15,29)(16,33)(18,23)(25,36);(2,4)(5,7)(6,11)(8,16)(9,23)(10,17)(12,25)(13,29)(14,26)(15,21)(18,22)(19,33)(20,35)(24,34)(27,36)(28,31)(30,32);(5,8,10,19)(6,12,14,27)(7,16,17,33)(9,24,22,20)(11,25,26,36)(13,30,21,28)(15,31,29,32)(18,35,23,34)Polycyclic presentation:
2^8|a^2=h,d^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 | #139737, #139751, #144376, … |
| maximal normal subgroup of | #139737, #139751, #144376, … |
| maximal quotient of | #245172, #245179, #245184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16904);