Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22)(23,30)(24,27)(25,29)(26,32)(28,31)(33,34)(35,36);(1,3,4,2)(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,22)(23,26)(24,28)(25,29)(27,31)(30,32)(33,35)(34,36);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,23,31,36,25,33,24,32)(22,26,27,34,29,35,28,30);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(21,24,25,31)(22,27,29,28)(23,32,33,36)(26,34,35,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=efh,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #139132, #139133, #139134, … |
| maximal normal subgroup of | #139132, #139133, #139134, … |
| maximal quotient of | #219266, #219270, #219292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15242);