Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7540 |
| automorphism | Aut(G) = #72286351 (order 2048) · Inn(G) = #1415 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1415 |
| commutator | C32 | C2×C4 |
| Frattini, cosocle | C2×C32 | C22 |
| prime radical | #2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(2,8,9,30)(3,13)(4,19)(5,21)(6,25)(10,36,34,44)(11,38,35,43)(12,39,27,33)(14,20,31,29)(15,23)(16,48)(17,52)(18,28)(22,47)(24,32,60,62)(26,58)(37,49,56,64)(40,41)(42,50)(45,59,53,57)(46,61)(51,63)(54,55);(1,2)(3,14)(4,20)(5,24)(6,27)(7,9)(8,28)(10,19)(11,21)(12,25)(13,30)(15,34)(16,49)(17,53)(18,31)(22,59)(23,29)(26,60)(32,63)(33,42)(35,58)(36,61)(37,47)(38,46)(39,52)(40,44)(41,43)(45,50)(48,62)(51,64)(54,57)(55,56)Polycyclic presentation:
2^8|a^2=d,c^2=efh,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=ceh,d^b=dh,e^b=ef,f^b=fg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7540, #153562, #153597, … |
| maximal normal subgroup of | C2×#7540, #153562, #153597, … |
| maximal quotient of | #94609, #94610, #94690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 528);