Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = C2×#72235851 (order 2048) · Inn(G) = C2×D4 · Out(G) = C2×#525 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | #121 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,12,17,13,18,22,30,14,16,23,28,24,29,34,38)(11,15,20,25,21,26,33,35,19,27,31,36,32,37,39,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,27)(17,25)(18,26)(22,33)(23,31)(24,32)(28,36)(29,37)(30,35)(34,39)(38,40);(2,6)(5,8)(10,16)(11,19)(17,28)(18,29)(20,31)(21,32)(30,38)(33,39)Polycyclic presentation:
2^8|a^2=e,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,c^b=cfh,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | #129656, #129661, #129672, … |
| maximal normal subgroup of | #129656, #129661, #129672, … |
| maximal quotient of | #124293, #124331, #124335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4612);