Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500160 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24033 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61089 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2931 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(4,17)(5,20)(6,19)(9,12)(10,26)(11,23)(13,14)(15,30)(18,31)(21,24)(22,27)(25,32)(28,29);(2,9)(4,17)(5,20)(6,19)(8,21)(10,25)(11,24)(12,23)(13,14)(15,30)(18,31)(22,32)(26,28)(27,29);(1,2,6,21,7,11,19,12)(3,10,15,32,16,27,30,28)(4,9,17,23,20,24,5,8)(13,29,14,22,18,25,31,26);(1,3)(2,10)(4,18)(5,14)(6,15)(7,16)(8,22)(9,25)(11,27)(12,28)(13,20)(17,31)(19,30)(21,32)(23,26)(24,29)Polycyclic presentation:
2^9|c^2=eh,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, #72231705, #72231714, … |
| maximal normal subgroup of | #72213945, #72231705, #72231714, … |
| maximal quotient of | #72226822, #72239893, #72250533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407356);