Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12301 |
| automorphism | Aut(G) = #72308067 (order 2048) · Inn(G) = #1613 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1613 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 155 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:
(1,2,7,12,8,14,6,13)(3,15,19,36,20,44,18,43)(4,21,23,10,24,39,29,35)(5,25,27,11,28,40,30,38)(9,26,41,17,33,49,32,50)(16,45,47,42,48,63,51,60)(22,37,56,53,57,34,55,58)(31,59,52,54,62,46,61,64);(1,3,8,20)(2,9,14,33)(4,16,24,48)(5,26,28,49)(6,19,7,18)(10,31,35,62)(11,36,38,43)(12,32,13,41)(15,25,44,40)(17,27,50,30)(21,52,39,61)(22,54,57,64)(23,51,29,47)(34,45,37,63)(42,58,60,53)(46,56,59,55);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,31)(11,37)(12,39)(13,21)(14,35)(15,42)(17,46)(18,51)(19,47)(20,48)(25,58)(26,54)(27,56)(28,57)(30,55)(32,61)(33,62)(34,38)(36,63)(40,53)(41,52)(43,45)(44,60)(49,64)(50,59)Polycyclic presentation:
2^8|a^2=fg,b^2=h,d^2=f,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gh,d^b=dfh,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 | #151238, #151239, #151242, … |
| maximal normal subgroup of | #151238, #151239, #151242, … |
| maximal quotient of | #126285, #126440, #126444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5290);