Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
855 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 855 conjugacy classes of subgroups — too many to draw. Only its 239 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:
(2,4)(5,6,10,14,11,16,9,15)(7,17,19,12,21,27,20,28)(8,22,23,29,25,13,24,30)(18,31,34,35,33,36,32,26);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,26)(14,17)(15,27)(16,28)(22,35)(23,32)(24,34)(25,33)(29,31)(30,36);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,30)(16,22)(17,31)(19,32)(20,34)(21,33)(27,36)(28,35);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,25,24)(12,17,28,27)(13,29,22,30)(18,32,33,34)(26,31,35,36)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147805, #147806, #147839, … |
| maximal normal subgroup of | #147805, #147806, #147839, … |
| maximal quotient of | #193791, #193800, #193804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13943);