Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3060 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
443 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(5,6,9,12,11,16,10,15)(7,17,18,13,21,28,20,29)(8,22,24,14,23,26,25,30)(19,27,32,35,34,36,33,31);(1,2)(3,4)(6,12,16,15)(7,18,21,20)(8,23)(14,22,30,26)(19,33,34,32)(24,25)(27,36)(31,35);(2,4)(5,7)(6,13)(8,19)(9,20)(10,18)(11,21)(12,17)(14,27)(15,28)(16,29)(22,35)(23,34)(24,33)(25,32)(26,31)(30,36);(5,8,10,25,11,23,9,24)(6,14,12,26,16,30,15,22)(7,19,18,32,21,34,20,33)(13,27,17,31,29,36,28,35)Polycyclic presentation:
2^8|a^2=fh,b^2=ef,d^2=eh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #502214, #504845, #504887, … |
| maximal normal subgroup of | #502214, #504845, #504887, … |
| maximal quotient of | #288645, #288648, #288651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18433);