Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
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 | #532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 79 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)(3,5)(4,7)(6,8)(9,10,12,17,14,18,25,32,13,16,24,29,26,30,37,39)(11,19,21,31,22,33,35,40,20,15,23,27,34,28,36,38);(2,5)(3,8)(6,7)(9,11)(10,15)(12,23)(13,20)(14,22)(16,19)(17,31)(18,28)(21,24)(25,36)(26,34)(27,29)(30,33)(32,40)(35,37)(38,39);(10,16)(11,20)(17,29)(18,30)(21,23)(22,34)(32,39)(35,36)Polycyclic presentation:
2^8|a^2=e,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 | #128703, #128718, #129499, … |
| maximal normal subgroup of | #128703, #128718, #129499, … |
| maximal quotient of | #122504, #122505, #122506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4576);