Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,4,9,5,10,16,23)(3,8,12,22,13,21,27,31)(6,7,15,19,17,20,28,30)(11,24,25,32,14,18,26,29);(2,7)(3,11)(8,21)(10,20)(12,25)(13,14)(18,24)(22,32)(26,27)(29,31);(1,3,6,14)(2,8,7,18)(4,12,15,26)(5,13,17,11)(9,22,19,29)(10,21,20,24)(16,27,28,25)(23,31,30,32)Polycyclic presentation:
2^8|a^2=e,c^2=h,e^2=g,b^a=bd,d^a=df,f^a=fh,c^b=cgh,e^b=ef,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132853, #132855, #132857, … |
| maximal normal subgroup of | #132853, #132855, #132857, … |
| maximal quotient of | #125073, #125089, #125107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4534);