Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9443 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | |
264 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 264 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:
(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18)(19,22,26,29)(20,23)(21,24)(25,28,31,32)(27,30);(1,3)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16)(17,19)(18,22,24,29)(20,25)(21,26)(23,28,30,32)(27,31);(2,8)(4,5)(7,15)(11,13)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32)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=egh,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132870, #132872, #132874, … |
| maximal normal subgroup of | #132870, #132872, #132874, … |
| maximal quotient of | #100125, #100129, #100141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2449);