Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = C4×#7622880 (order 2048) · Inn(G) = C22⋊C4 · Out(G) = C22×C4×D4 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C22⋊C4 |
| commutator | C22 | C2×C4×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | C8×C22⋊C4 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
205 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(1,2,5,10,6,12,21,32,7,11,22,31,23,33,45,54)(3,13,16,35,17,36,41,57,15,8,24,26,40,27,47,50)(4,9,18,28,19,30,42,52,20,29,43,51,44,53,60,63)(14,34,38,55,39,56,59,64,37,25,46,48,58,49,61,62);(1,3)(2,8,11,13)(4,14)(5,16)(6,17)(7,15)(9,25,29,34)(10,26,31,35)(12,27,33,36)(18,38)(19,39)(20,37)(21,41)(22,24)(23,40)(28,48,51,55)(30,49,53,56)(32,50,54,57)(42,59)(43,46)(44,58)(45,47)(52,62,63,64)(60,61);(1,4,5,18,6,19,21,42,7,20,22,43,23,44,45,60)(2,9,10,28,12,30,32,52,11,29,31,51,33,53,54,63)(3,14,16,38,17,39,41,59,15,37,24,46,40,58,47,61)(8,25,26,48,27,49,50,62,13,34,35,55,36,56,57,64)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,e^2=g,g^2=h,b^a=bd,f^a=fh,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #94126, #94127, #128461, … |
| maximal normal subgroup of | #94126, #94127, #128461, … |
| maximal quotient of | #128248, #128265, #128284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5601);