Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500168 |
| 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 | #61079 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2932 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,10)(5,20)(6,24)(7,12)(8,16)(9,22)(11,21)(14,25)(15,29)(17,27)(18,28)(19,23)(26,31)(30,32);(1,2)(3,14)(4,19)(5,9)(6,24)(7,12)(8,17)(10,23)(11,21)(13,18)(15,28)(16,30)(20,22)(25,29)(26,32)(27,31);(1,3,6,25,7,15,21,18)(2,8,11,30,12,26,24,27)(4,17,22,31,23,32,5,16)(9,28,10,29,20,14,19,13);(1,2)(3,8)(4,20)(5,10)(6,11)(7,12)(9,23)(13,31)(14,17)(15,26)(16,29)(18,27)(19,22)(21,24)(25,30)(28,32)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=gh,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72231700, #72231714, … |
| maximal normal subgroup of | #72213937, #72231700, #72231714, … |
| maximal quotient of | #72231480, C2×#500168, #72239865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407364);