Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18203 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| prime radical | #62331 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
689 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 689 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 32 points:
(1,2,7,13,8,15,6,14)(3,16,19,10,20,32,27,29)(4,21,18,11,23,28,22,31)(5,24,17,12,26,9,25,30);(2,9)(4,18)(5,25)(6,7)(10,28)(11,32)(12,13)(14,30)(15,24)(16,31)(17,26)(19,27)(21,29)(22,23);(1,3,8,20)(2,10,15,29)(4,17,23,25)(5,18,26,22)(6,27,7,19)(9,28,24,21)(11,30,31,12)(13,32,14,16);(1,4,7,18,8,23,6,22)(2,11,14,21,15,31,13,28)(3,17,19,26,20,25,27,5)(9,29,12,32,24,10,30,16)Polycyclic presentation:
2^9|a^2=gh,c^2=i,d^2=g,e^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,h^a=hi,d^b=dg,e^b=egi,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231599, #72231714, #72231715, … |
| maximal normal subgroup of | #72231599, #72231714, #72231715, … |
| maximal quotient of | #72250528, #72250959, C2×#486697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393893);