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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #418 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1824 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,8,11,7,30)(3,14,17,35,18,9,29,34)(4,19,22,51,20,52,31,41)(5,23,26,39,28,55,27,12)(10,25,36,53,38,59,37,21)(15,43,46,64,44,60,47,62)(16,48,50,40,24,56,58,33)(32,49,61,45,42,57,63,54);(1,3)(2,9)(4,15)(5,24)(6,29)(7,17)(8,18)(10,32)(11,14)(12,40)(13,35)(16,28)(19,43)(20,44)(21,54)(22,47)(23,48)(25,57)(26,50)(27,58)(30,34)(31,46)(33,39)(36,63)(37,61)(38,42)(41,64)(45,53)(49,59)(51,62)(52,60)(55,56);(1,4,7,31,8,20,6,22)(2,10,30,37,11,38,13,36)(3,15,17,46,18,44,29,47)(5,25,27,21,28,59,26,53)(9,32,34,61,14,42,35,63)(12,41,55,52,39,51,23,19)(16,49,50,45,24,57,58,54)(33,62,48,43,40,64,56,60)Polycyclic presentation:
2^8|a^2=f,c^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139577, #139578, #139581, … |
| maximal normal subgroup of | #139577, #139578, #139581, … |
| maximal quotient of | #135269, #135289, #135349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5866);