Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | Aut(G) = AGL(1,ℤ/8)×AGL(1,ℤ/16) (order 4096) · Inn(G) = D4×D8 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
318 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 105 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 — 4 generators on 64 points:
(1,2,9,15)(3,16,21,51)(4,22,11,37)(5,26,29,25)(6,30,35,46)(7,14,8,13)(10,38,42,19)(12,33,47,34)(17,18,23,56)(20,58,36,44)(24,54,43,28)(27,61,39,40)(31,48,63,41)(32,45,64,62)(49,50,55,53)(52,57,60,59);(1,3,9,21)(2,10,15,42)(4,23,11,17)(5,27,29,39)(6,31,35,63)(7,20,8,36)(12,44,47,58)(13,41,14,48)(16,46,51,30)(18,54,56,28)(19,33,38,34)(22,59,37,57)(24,61,43,40)(25,50,26,53)(32,55,64,49)(45,52,62,60);(1,4,9,11)(2,5,15,29)(3,17,21,23)(6,24,35,43)(7,26,8,25)(10,39,42,27)(12,28,47,54)(13,37,14,22)(16,49,51,55)(18,44,56,58)(19,52,38,60)(20,50,36,53)(30,32,46,64)(31,40,63,61)(33,45,34,62)(41,59,48,57);(1,5)(2,11)(3,18)(4,15)(6,32)(7,22)(8,37)(9,29)(10,40)(12,45)(13,26)(14,25)(16,50)(17,44)(19,57)(20,55)(21,56)(23,58)(24,46)(27,31)(28,34)(30,43)(33,54)(35,64)(36,49)(38,59)(39,63)(41,52)(42,61)(47,62)(48,60)(51,53)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gh,c^b=ch,d^b=de,e^b=eg,g^b=gh,f^c=fh,e^d=eg,g^d=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #151525, #151527, #151581, … |
| maximal normal subgroup of | #151525, #151527, #151581, … |
| maximal quotient of | #491739, #491740, #491754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26848);