Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12170 |
| automorphism | Aut(G) = #72308439 (order 2048) · Inn(G) = #1598 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1598 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | #575 | C22 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 35 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)(5,22)(6,28)(7,12)(8,15)(9,20)(10,27)(11,25)(14,41)(16,48)(17,40)(19,52)(21,34)(23,37)(24,57)(26,38)(29,35)(30,46)(31,47)(32,59)(33,60)(36,45)(39,43)(42,49)(44,62)(50,55)(51,54)(53,61)(56,64)(58,63);(1,3,7,17)(2,8,12,31)(4,14,20,43)(5,23,26,59)(6,29,25,16)(9,30,18,49)(10,35,36,48)(11,37,28,32)(13,38,40,22)(15,45,47,27)(19,53,24,58)(21,56,54,44)(33,62,50,64)(34,63,51,61)(39,52,41,57)(42,60,46,55);(1,4,7,20)(2,9)(3,14,17,43)(5,24,26,19)(6,21,25,54)(8,30)(10,33)(11,34)(12,18)(13,39)(15,46,47,42)(16,44,29,56)(22,57)(23,58,59,53)(27,60,45,55)(28,51)(31,49)(32,61)(35,62)(36,50)(37,63)(38,52)(40,41)(48,64)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=fh,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gh,d^b=df,e^b=egh,f^b=fh,d^c=dg,e^c=eh,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12171, #484651, #484653, … |
| maximal normal subgroup of | C2×#12171, #484651, #484653, … |
| maximal quotient of | #123172, #123174, #123180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5159);