Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9385 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
263 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18)(19,22,26,31)(20,27,29,23)(21,24)(25,30)(28,32);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19)(18,22,24,31)(20,28)(21,26)(23,32,27,30)(25,29);(3,11)(7,15)(10,16)(12,13)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132839, #132840, #132843, … |
| maximal normal subgroup of | #132839, #132840, #132843, … |
| maximal quotient of | #100068, #100070, #100076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2400);