Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#121 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
961 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 961 conjugacy classes of subgroups — too many to draw. Only its 183 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:
(1,2)(3,4)(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,20)(13,25)(18,33)(19,29)(21,28)(23,30)(24,32)(26,31)(27,35)(34,36);(2,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,17)(15,28)(16,29)(18,24)(22,33)(25,35)(26,34)(27,31)(32,36);(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,22)(15,31)(16,32)(17,30)(19,34)(20,23)(21,35)(28,36)(29,33);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,24,26,25)(12,17,29,28)(13,22,32,31)(18,34,35,23)(27,30,33,36)Polycyclic presentation:
2^8|d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,g^b=gh,e^c=eh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137146, #137153, #142242, … |
| maximal normal subgroup of | #137146, #137153, #142242, … |
| maximal quotient of | #232436, #232440, #232441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16426);