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 | #62328 | 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 | |
725 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 725 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)(4,21,22,11,23,30,18,31)(5,24,17,12,26,9,25,28);(2,9)(4,22)(5,25)(6,7)(10,21)(11,16)(12,13)(14,28)(15,24)(17,26)(18,23)(19,27)(29,30)(31,32);(1,3)(2,10)(4,17)(5,18)(6,27)(7,19)(8,20)(9,21)(11,28)(12,31)(13,32)(14,16)(15,29)(22,26)(23,25)(24,30);(1,4,6,18,8,23,7,22)(2,11,13,30,15,31,14,21)(3,17,27,5,20,25,19,26)(9,10,28,32,24,29,12,16)Polycyclic presentation:
2^9|a^2=gh,d^2=gi,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 | #72231593, #72231705, #72231715, … |
| maximal normal subgroup of | #72231593, #72231705, #72231715, … |
| maximal quotient of | #72250471, #72250929, C2×#486698, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393894);