Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8412 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #379 · Out(G) = C25 |
| 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 | #3515 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
352 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6,7,8)(9,10)(11,19,22,15)(12,23,25,32)(13,18)(14,17)(16,24,30,26)(20,35)(21,34,39,29)(27,31)(28,36)(33,38)(37,40);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,28)(17,19)(18,29)(23,40)(24,36)(25,37)(26,38)(27,39)(30,33)(31,34)(32,35);(1,3,5,4)(2,6,7,8)(9,12,14,25)(10,16,17,30)(11,20,22,37)(13,26,27,24)(15,28,19,33)(18,32,31,23)(21,38,39,36)(29,35,34,40)Polycyclic presentation:
2^8|a^2=f,c^2=gh,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fgh,e^c=eg,f^c=fg43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8415, #186123, #186125, … |
| maximal normal subgroup of | C2×#8415, #186123, #186125, … |
| maximal quotient of | #97165, #97167, #97171, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1403);