Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512779 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2177 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | #33981 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8,10,31)(3,13)(4,17)(5,23)(6,25)(9,34,35,61)(11,40,39,56)(12,41,27,33)(14,43)(15,47)(16,46)(18,24,32,30)(19,57)(20,26)(21,59)(22,28)(29,42,36,38)(37,55,62,48)(44,51)(49,54)(52,63,60,64)(53,58);(1,2)(3,9)(4,18)(5,24)(6,27)(7,10)(8,28)(11,23)(12,25)(13,42)(14,35)(15,48)(16,52)(17,31)(19,55)(20,39)(21,40)(22,32)(26,30)(29,51)(33,54)(34,57)(36,53)(37,45)(38,43)(41,59)(44,64)(46,60)(47,61)(49,56)(50,62)(58,63);(1,3,6,13,7,14,25,43)(2,9,27,42,10,35,12,38)(4,19,21,58,22,15,49,44)(5,16,23,45,26,46,20,50)(8,29,41,61,31,36,33,34)(11,37,30,60,39,62,24,52)(17,53,54,57,28,51,59,47)(18,55,40,63,32,48,56,64);(3,14)(9,35)(13,43)(15,19)(16,46)(29,36)(34,61)(37,62)(38,42)(44,58)(45,50)(47,57)(48,55)(51,53)(52,60)(63,64)Polycyclic presentation:
2^9|a^2=f,c^2=fh,e^2=ghi,g^2=hi,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gh,h^a=hi,e^b=egi,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224987, #72225005, #72225006, … |
| maximal normal subgroup of | #72224987, #72225005, #72225006, … |
| maximal quotient of | #72232010, #72232011, #72232019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419975);