Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13667 |
| automorphism | Aut(G) = #72285396 (order 2048) · Inn(G) = #2184 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | #3414 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 37 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:
(1,2,7,12)(3,13,15,42)(4,18,20,48)(5,24,27,58)(6,28,26,11)(8,30,31,14)(9,34,36,22)(10,38,40,21)(16,46,45,63)(17,47,29,43)(19,39,51,64)(23,56,54,50)(25,35,59,62)(32,57,60,53)(33,44,41,52)(37,55,49,61);(1,3)(2,8)(4,19)(5,25)(6,29)(7,15)(9,35)(10,39)(11,41)(12,31)(13,18)(14,22)(16,21)(17,26)(20,51)(23,57)(24,32)(27,59)(28,33)(30,34)(36,62)(37,63)(38,45)(40,64)(42,48)(43,56)(44,61)(46,49)(47,50)(52,55)(53,54)(58,60);(4,20)(9,36)(13,42)(14,30)(18,48)(19,51)(22,34)(23,54)(35,62)(37,49)(43,47)(44,52)(46,63)(50,56)(53,57)(55,61)Polycyclic presentation:
2^8|a^2=h,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,d^b=df,e^b=eh,f^b=fg,g^b=gh,d^c=dh31 conjugacy classes — view the full character table.
| maximal subgroup of | #151183, #151184, #151186, … |
| maximal normal subgroup of | #151183, #151184, #151186, … |
| maximal quotient of | #151173, #151174, #151175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6656);