Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3497 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
665 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 665 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(2,3)(5,6)(7,11)(8,18)(9,21)(10,14)(12,20)(13,24)(15,32)(16,30)(17,27)(19,31)(22,26)(23,29)(25,34)(28,35)(33,36);(1,2,4,3)(5,7,10,17)(6,11,14,27)(8,15,19,33)(9,22,24,16)(12,28,29,25)(13,30,21,26)(18,32,31,36)(20,35,23,34);(5,8,10,19)(6,12,14,29)(7,15,17,33)(9,23,24,20)(11,25,27,28)(13,31,21,18)(16,35,22,34)(26,36,30,32);(1,3,4,2)(8,19)(12,29)(15,33)(18,31)(20,23)(25,28)(32,36)(34,35)Polycyclic presentation:
2^8|b^2=eh,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #245726, #245733, #245747, … |
| maximal normal subgroup of | #245726, #245733, #245747, … |
| maximal quotient of | #261088, #261091, #261096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17217);