Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) = C2×#72215858 (order 2048) · Inn(G) = #120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #379 |
| prime radical | #3529 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
364 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 71 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:
(2,5,6,7)(9,10)(11,19,22,15)(12,23,24,34)(13,18)(14,17)(16,25,31,27)(20,36)(21,33,26,30)(28,32)(29,38)(35,39)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,20)(13,26)(14,22)(16,29)(17,19)(18,33)(21,28)(23,36)(24,37)(25,39)(27,38)(30,32)(31,35)(34,40);(3,8)(5,7)(9,12)(10,16)(11,20)(13,27)(14,24)(15,29)(17,31)(18,34)(19,35)(21,39)(22,37)(23,32)(25,28)(26,38)(30,36)(33,40)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #152553, #152560, #153131, … |
| maximal normal subgroup of | #152553, #152560, #153131, … |
| maximal quotient of | #97000, #97001, #97002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1492);