Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13215 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1961 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
217 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 217 conjugacy classes of subgroups — too many to draw. Only its 65 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 — 3 generators on 64 points:
(1,2)(3,14)(4,20,24,53)(5,26)(6,12)(7,30)(8,13)(9,17)(10,23,38,25)(11,27)(15,39)(16,46,48,62)(18,41)(19,43)(21,54,57,40)(22,58,59,37)(28,31)(29,42)(32,56,51,63)(33,49)(34,44)(35,50)(36,47,61,64)(45,60,55,52);(1,3,15,44,7,18,5,17,8,19,27,49,6,29,28,50)(2,9,31,41,12,34,11,14,13,35,26,42,30,33,39,43)(4,21,55,64,25,48,59,63,24,57,45,47,23,16,22,56)(10,36,52,54,20,51,58,46,38,61,60,40,53,32,37,62);(1,4)(2,10,13,38)(3,16,19,48)(5,22)(6,23)(7,25)(8,24)(9,32)(11,37,39,58)(12,20,30,53)(14,40)(15,45)(17,47,50,64)(18,21,29,57)(26,52,31,60)(27,55)(28,59)(33,61)(34,36)(35,51)(41,46)(42,62)(43,54)(44,63,49,56)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=eg,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=cg,e^b=eh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13216, #495144, #495574, … |
| maximal normal subgroup of | C2×#13216, #495144, #495574, … |
| maximal quotient of | #140557, #140558, #140598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6204);