Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13377 |
| commutator, Frattini, cosocle | #532 | C23 |
| prime radical | #3549 | C22 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,8,11,7,15,4,14)(3,16,18,9,21,25,22,27)(5,12,24,13,6,10,23,29)(17,31,19,30,20,26,32,28);(1,3,8,18,7,21,4,22)(2,9,11,25,15,27,14,16)(5,17,24,19,6,20,23,32)(10,28,29,31,12,30,13,26);(2,10)(3,17)(4,8)(5,6)(9,25)(11,13)(12,15)(14,29)(16,27)(18,32)(19,22)(20,21)(26,30)(28,31)Polycyclic presentation:
2^9|a^2=dh,b^2=dh,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fgh,h^a=hi,c^b=cf,d^b=di,e^b=eg,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi,g^c=gh,f^e=fhi,g^e=gi,g^f=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72339674, #72339806, #72339807 |
| maximal normal subgroup of | #72339674, #72339806, #72339807 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60426);