Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | Aut(G) = C22×#648360 (order 2048) · Inn(G) = #379 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3547 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
336 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 336 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) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,10,12,6)(4,9)(5,8)(7,11,15,13)(14,16)(17,18,22,27,23,28,21,26)(19,29,33,24,32,39,31,38)(20,25,34,40,35,41,36,42)(30,44,45,48,46,43,47,37);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,14)(15,16)(17,19)(18,24)(20,30)(21,33)(22,31)(23,32)(25,37)(26,39)(27,29)(28,38)(34,47)(35,46)(36,45)(40,43)(41,48)(42,44);(1,3)(2,7)(4,13)(5,12)(6,9)(8,15)(10,16)(11,14)(17,20,23,35)(18,25,28,41)(19,30,32,46)(21,36,22,34)(24,37,38,48)(26,42,27,40)(29,43,39,44)(31,47,33,45)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=eg,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8502, #187508, #187509, … |
| maximal normal subgroup of | C2×#8502, #187508, #187509, … |
| maximal quotient of | #96982, #96985, #96986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1490);