Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486689 |
| 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 | #62270 | 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 | |
727 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 727 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:
(1,2,7,11,6,12,5,21)(3,13,22,25,16,28,14,9)(4,17,18,26,19,8,20,10)(15,30,32,23,31,24,29,27);(2,8)(3,14)(4,18)(5,7)(9,24)(10,21)(11,26)(12,17)(13,23)(16,22)(19,20)(25,30)(27,28)(29,32);(4,19)(15,31)(18,20)(29,32);(1,3,7,22,6,16,5,14)(2,9,21,28,12,25,11,13)(4,15,18,32,19,31,20,29)(8,23,26,30,17,27,10,24)Polycyclic presentation:
2^9|a^2=gh,d^2=gi,e^2=g,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=egi,f^b=fhi,g^b=gi,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231596, #72231705, #72231710, … |
| maximal normal subgroup of | #72231596, #72231705, #72231710, … |
| maximal quotient of | #72250377, C2×#486691, #72265452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393887);