Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | Aut(G) = C23×#2846 (order 1024) · Inn(G) = #379 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #538 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
315 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,5,11,27,34,57,60)(3,13,16,42,15,48,25,56)(4,19,22,51,21,37,53,62)(6,9,26,33,7,12,28,40)(8,29,32,14,44,46,39,45)(10,35,38,24,41,55,64,23)(17,43,49,63,18,36,50,61)(20,30,52,58,47,31,54,59);(1,3,6,17)(2,8,9,30)(4,20,23,45)(5,25,26,50)(7,18,27,15)(10,36,19,48)(11,39,33,59)(12,31,34,44)(13,41,43,37)(14,21,47,24)(16,28,49,57)(22,54,35,46)(29,53,52,55)(32,40,58,60)(38,63,51,42)(56,64,61,62);(2,9)(4,21)(5,26)(8,30)(10,37)(12,34)(13,36)(14,45)(16,49)(19,41)(20,47)(22,55)(23,24)(25,50)(28,57)(29,54)(31,44)(35,53)(38,64)(42,56)(43,48)(46,52)(51,62)(61,63)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,f^c=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8490, #187344, #187345, … |
| maximal normal subgroup of | C2×#8490, #187344, #187345, … |
| maximal quotient of | #97205, #97207, #97208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1478);