Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #102 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C2 | C2×D4⋊C8 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
190 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,7,12,22,30,8,13,21,10,6,23,4,19)(3,14,17,9,18,26,20,28,15,31,16,25,24,29,32,27);(1,3)(2,9)(4,20)(5,16)(6,24)(7,18)(8,15)(10,26)(11,29)(12,28)(13,25)(14,30)(17,21)(19,31)(22,32)(23,27);(3,15)(9,25)(14,31)(16,17)(18,24)(20,32)(26,29)(27,28)Polycyclic presentation:
2^8|a^2=e,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=dfh,e^b=eh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128554, #128584, #128675, … |
| maximal normal subgroup of | #128554, #128584, #128675, … |
| maximal quotient of | #122465, #122466, #122467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4592);