Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = #120×#525 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | #121 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 159 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:
(1,2,5,8,6,3,4,7)(9,10,13,19,14,20,28,37,15,17,29,33,12,18,26,35)(11,21,24,38,23,32,39,36,22,34,27,40,25,16,30,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,25)(13,27)(14,23)(15,22)(17,32)(18,34)(19,36)(20,21)(24,29)(26,39)(28,30)(31,33)(35,38)(37,40);(1,4,6,5)(2,7,3,8)(10,17)(11,22)(18,20)(19,33)(23,25)(24,27)(30,39)(35,37)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=fh,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,c^b=ch,d^b=dfh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128654, #128657, #129661, … |
| maximal normal subgroup of | #128654, #128657, #129661, … |
| maximal quotient of | #124102, #124133, #124136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4608);