Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| 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 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 409 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:
(5,6)(7,17)(8,22,26,31)(9,15)(10,14)(11,16)(12,19)(13,24,32,25)(18,27,35,36)(20,29)(21,28)(23,33,34,30);(1,2)(3,4)(5,7,11,21)(6,12,16,29)(8,23,26,34)(9,20,10,19)(13,30,32,33)(14,28,15,17)(18,24,35,25)(22,27,31,36);(2,4)(5,8,9,24,11,26,10,25)(6,13,14,31,16,32,15,22)(7,18,19,34,21,35,20,23)(12,27,17,33,29,36,28,30)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11184, #357939, #358177, … |
| maximal normal subgroup of | C2×#11184, #357939, #358177, … |
| maximal quotient of | #118687, #118772, #118778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4172);