Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140287 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9572 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(2,7,10,16)(3,12)(4,8)(5,19)(9,25,27,31)(11,22,28,21)(13,20,32,26)(14,17)(15,30)(18,24);(1,2,8,22,19,28,4,16,6,10,24,21,5,11,18,7)(3,13,29,26,30,9,14,25,15,32,23,20,12,27,17,31);(1,3,4,17,5,12,8,23,6,15,18,14,19,30,24,29)(2,9,7,20,11,13,21,25,10,27,16,26,28,32,22,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=dh,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cg,d^b=di,f^b=fi,d^c=di,e^c=egh,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72214066, #72214217, #72214308 |
| maximal normal subgroup of | #72214066, #72214217, #72214308 |
| maximal quotient of | #72216476, #72227261, #72227264 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47483);