Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486340 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18194 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
887 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 887 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)(3,16)(4,20)(5,9)(6,14)(7,13)(8,15)(10,18)(11,21)(12,23)(17,31)(19,27)(22,32)(24,29)(25,30)(26,28);(2,9)(4,21)(5,23)(6,7)(10,11)(12,14)(13,28)(15,29)(16,31)(17,22)(18,25)(20,27)(24,26)(30,32);(1,3,8,19)(2,10,15,30)(4,5,17,24)(6,25,7,18)(9,11,29,32)(12,31,28,20)(13,27,14,16)(21,23,22,26);(1,4,7,21,8,17,6,22)(2,11,14,20,15,32,13,31)(3,5,18,23,19,24,25,26)(9,27,28,10,29,16,12,30)Polycyclic presentation:
2^9|c^2=i,d^2=g,e^2=gi,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=eg,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 | #72231620, #72231708, #72231715, … |
| maximal normal subgroup of | #72231620, #72231708, #72231715, … |
| maximal quotient of | #72250550, #72250968, C2×#486341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393537);