Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12741 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1786 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 439 conjugacy classes of subgroups — too many to draw. Only its 85 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)(3,13)(4,18,20,9)(5,22)(6,27)(7,11)(8,15)(10,25)(12,24)(14,40,42,38)(16,46)(17,39)(19,50,53,56)(21,33,52,49)(23,36)(26,37)(28,34)(29,41,47,63)(30,45)(31,57)(32,54,48,62)(35,44)(43,59,55,61)(51,58,64,60);(1,3,7,17)(2,8,11,30)(4,14,20,42)(5,23,26,57)(6,28,24,16)(9,29,18,47)(10,34,35,46)(12,36,27,31)(13,37,39,22)(15,44,45,25)(19,51,53,64)(21,55,52,43)(32,59,48,61)(33,60,49,58)(38,50,40,56)(41,62,63,54);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,29)(10,32)(11,18)(12,33)(13,38)(15,41)(16,43)(17,42)(22,56)(23,51)(24,52)(25,54)(26,53)(27,49)(28,55)(30,47)(31,58)(34,59)(35,48)(36,60)(37,50)(39,40)(44,62)(45,63)(46,61)(57,64)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130777, #130778, #130781, … |
| maximal normal subgroup of | #130777, #130778, #130781, … |
| maximal quotient of | #129518, #129519, #129561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5730);