Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
397 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,14,9,26)(7,16,19,32,20,12,18,28)(8,21,24,30,25,31,23,13)(17,29,22,27,35,36,34,33);(5,7,11,20)(6,12,14,16)(8,22,25,34)(9,19,10,18)(13,29,30,36)(15,32,26,28)(17,23,35,24)(21,33,31,27);(2,4)(5,8)(6,13,14,30)(7,17)(9,23)(10,24)(11,25)(12,27,16,33)(15,21,26,31)(18,34)(19,22)(20,35)(28,36,32,29)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=d,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,g^b=gh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11188, #357945, C4×#2049, … |
| maximal normal subgroup of | C2×#11188, #357945, C4×#2049, … |
| maximal quotient of | #118689, #118784, #118788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4176);