Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | Aut(G) = #72313012 (order 4096) · Inn(G) = #418 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #418 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1786 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
445 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 445 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)(6,28)(7,11)(8,15)(10,26)(12,25)(14,39,42,60)(16,40)(17,46)(19,49,24,57)(21,34,51,48)(23,38)(27,54)(29,35)(30,41,47,44)(31,56)(32,45)(33,52,36,58)(37,59)(43,62,53,64)(50,61,55,63);(1,3)(2,8)(4,14)(5,23)(6,29)(7,16)(9,30)(10,35)(11,32)(12,38)(13,22)(15,26)(17,25)(18,47)(19,50)(20,42)(21,53)(24,55)(27,56)(28,31)(33,62)(34,63)(36,64)(37,46)(39,49)(40,54)(41,52)(43,51)(44,58)(45,59)(48,61)(57,60);(1,4)(2,9)(3,14)(5,24)(6,21)(7,20)(8,30)(10,36)(11,18)(12,34)(13,39)(15,44)(16,42)(17,43)(19,27)(22,49)(23,55)(25,51)(26,58)(28,48)(29,53)(31,61)(32,47)(33,37)(35,64)(38,63)(40,60)(41,45)(46,62)(50,56)(52,59)(54,57)Polycyclic presentation:
2^8|a^2=f,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,e^b=eh,g^b=gh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139571, #139572, #139573, … |
| maximal normal subgroup of | #139571, #139572, #139573, … |
| maximal quotient of | #129507, #129558, #129775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5733);