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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
216 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(1,2,5,10,6,9,4,8)(3,7,11,14,12,15,13,16)(17,18,23,26,22,19,21,25)(20,24,30,32,29,27,28,31);(1,3,4,11,6,12,5,13)(2,7,10,16,9,15,8,14)(17,19,22,18)(20,27,29,24)(21,25,23,26)(28,31,30,32);(1,4,6,5)(2,8,9,10)(3,11,12,13)(7,14,15,16)(17,20,22,29)(18,24,19,27)(21,30,23,28)(25,32,26,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=h,d^2=gh,f^2=h,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 | #132614, #132616, #132618, … |
| maximal normal subgroup of | #132614, #132616, #132618, … |
| maximal quotient of | #100133, #100134, #100149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2444);