Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | Aut(G) = C4×#33553 (order 1024) · Inn(G) = #353 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #353 |
| commutator | C2×M16 | C2×C4 |
| Frattini, cosocle | #408 | C22 |
| socle | C2 | #1343 |
| prime radical | #2017 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
120 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,10,7,11,24,39)(3,13,15,42,17,43,47,62)(4,9,20,33,21,35,53,37)(6,26,25,56,19,12,40,38)(8,28,29,50,30,57,58,18)(14,31,44,36,45,41,46,61)(16,49,48,64,27,32,23,55)(22,54,52,63,51,34,60,59);(1,3)(2,8)(4,18)(5,23)(6,27)(7,17)(9,32)(10,36)(11,30)(12,41)(13,34)(14,22)(15,40)(16,19)(20,46)(21,50)(24,48)(25,47)(26,31)(28,52)(29,38)(33,42)(35,49)(37,62)(39,61)(43,54)(44,53)(45,51)(55,63)(56,58)(57,60)(59,64)Polycyclic presentation:
2^8|a^2=d,c^2=fh,d^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=def,f^b=fh,g^b=gh,d^c=dg,e^c=eh,e^d=eh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7407, #150195, #150344, … |
| maximal normal subgroup of | C2×#7407, #150195, #150344, … |
| maximal quotient of | #93207, #93208, #93209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 395);