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 | |
183 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)(3,13,16,8)(4,18)(5,22)(6,28)(7,10)(9,20)(11,26)(12,25)(14,41,42,60)(15,45,23,55)(17,32,47,40)(19,50)(21,53)(24,38)(27,54)(29,37)(30,43,39,52)(31,48,36,57)(33,58)(34,51)(35,59)(44,62,49,64)(46,61,56,63);(1,3)(2,8)(4,14)(5,23)(6,17)(7,16)(9,30)(10,13)(11,36)(12,32)(15,27)(18,41)(19,42)(20,52)(21,44)(22,45)(24,56)(25,47)(26,57)(28,40)(29,49)(31,35)(33,61)(34,39)(37,64)(38,63)(43,51)(46,58)(48,59)(50,60)(53,62)(54,55);(1,4,6,29,7,19,25,21)(2,9,28,33,10,34,12,38)(3,14,17,49,16,42,47,44)(5,24,26,20,27,58,59,51)(8,30,40,61,13,39,32,63)(11,37,54,50,35,53,22,18)(15,46,48,43,23,56,57,52)(31,62,45,41,36,64,55,60)Polycyclic presentation:
2^8|a^2=f,c^2=dg,e^2=gh,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, #139579, … |
| maximal normal subgroup of | #139577, #139578, #139579, … |
| maximal quotient of | #135269, #135289, #135348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5865);