Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| automorphism | Aut(G) order 4096 |
| 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 | C2×C42 | C23 |
| prime radical | #392 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
257 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 257 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,6,11,23,9,26,27)(3,14,17,20,10,15,28,19)(4,18,21,25,32,16,30,31)(5,12,22,29,7,13,8,24);(1,3,15,27,7,18,30,29)(2,8,25,4,13,6,17,19)(5,16,21,24,23,10,14,11)(9,22,31,32,12,26,28,20);(2,9)(3,16)(4,19)(6,8)(10,18)(11,29)(12,13)(17,28)(20,32)(22,26)(24,27)(25,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^b=eh,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132885, #132886, #132889, … |
| maximal normal subgroup of | #132885, #132886, #132889, … |
| maximal quotient of | #100165, #100166, #100167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2482);