Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = #72285904 (order 1024) · Inn(G) = C2×D4 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C2×D4 |
| commutator | C4 | C22×C16 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | C2×C22⋊C16 |
| prime radical | C2×#41 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
145 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 145 conjugacy classes of subgroups — too many to draw. Only its 77 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:
(1,2,5,12,6,13,24,39,7,14,25,40,27,41,52,61,8,10,26,33,28,34,53,57,4,11,21,36,22,37,49,59)(3,15,17,42,18,43,46,62,16,30,44,38,45,55,63,60,19,35,23,50,47,58,51,64,20,9,29,31,48,32,54,56);(1,3)(2,9)(4,20)(5,23)(6,18)(7,16)(8,19)(10,30)(11,35)(12,38)(13,32)(14,15)(17,26)(21,44)(22,48)(24,51)(25,29)(27,45)(28,47)(31,33)(34,55)(36,42)(37,58)(39,60)(40,50)(41,43)(46,53)(49,63)(52,54)(56,57)(59,62)(61,64);(2,10)(9,30)(11,14)(12,33)(13,34)(15,35)(31,38)(32,55)(36,40)(37,41)(39,57)(42,50)(43,58)(56,60)(59,61)(62,64)Polycyclic presentation:
2^8|a^2=e,d^2=h,e^2=f,f^2=g,g^2=h,b^a=bd,c^a=ch,d^b=dh,e^b=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | #151895, #151904, #152007, … |
| maximal normal subgroup of | #151895, #151904, #152007, … |
| maximal quotient of | #149521, #149522, #149603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6546);