Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13609 |
| automorphism | Aut(G) = C2×#493397 (order 1024) · Inn(G) = #2127 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2127 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4⋊Q8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
152 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 43 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:
(1,2,5,11,7,13,25,40)(3,14,16,38,18,31,45,57)(4,9,21,34,20,35,49,62)(6,12,27,39,28,10,24,37)(8,23,30,26,32,44,58,17)(15,41,42,60,19,46,47,63)(22,33,53,59,50,36,52,61)(29,54,55,48,56,51,64,43);(1,3,7,18)(2,8,13,32)(4,19,20,15)(5,23,25,44)(6,26,28,17)(9,29,35,56)(10,31,12,14)(11,38,40,57)(16,24,45,27)(21,51,49,54)(22,43,50,48)(30,39,58,37)(33,41,36,46)(34,60,62,63)(42,53,47,52)(55,59,64,61);(1,4)(2,9)(3,15)(5,21)(6,22)(7,20)(8,29)(10,36)(11,34)(12,33)(13,35)(14,41)(16,42)(17,43)(18,19)(23,54)(24,52)(25,49)(26,48)(27,53)(28,50)(30,55)(31,46)(32,56)(37,61)(38,60)(39,59)(40,62)(44,51)(45,47)(57,63)(58,64)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=gh,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=cg,d^b=dgh,e^b=efg,f^b=fh,d^c=dh,e^d=egh,f^e=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | #152140, #152141, #152142, … |
| maximal normal subgroup of | #152140, #152141, #152142, … |
| maximal quotient of | #149596, #149637, #149716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6600);