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) = C22 |
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 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
246 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(1,2,4,9,5,10,16,24)(3,11,13,23,12,25,26,32)(6,8,15,21,17,22,28,31)(7,18,20,27,19,29,30,14);(1,3,6,14)(2,7,8,11)(4,13,15,18)(5,12,17,27)(9,23,21,20)(10,19,22,25)(16,26,28,29)(24,32,31,30);(3,12)(7,19)(11,25)(13,26)(14,27)(18,29)(20,30)(23,32)Polycyclic presentation:
2^8|a^2=e,b^2=h,e^2=g,b^a=bd,d^a=df,f^a=fh,c^b=cg,e^b=ef,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132862, #132864, #132866, … |
| maximal normal subgroup of | #132862, #132864, #132866, … |
| maximal quotient of | #125070, #125086, #125104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4531);