Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366542 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3508 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8)(5,6)(9,21)(10,22)(11,12)(14,19)(16,25)(17,20)(23,31)(24,26)(27,32)(28,29)(33,34)(35,39)(36,46)(37,49)(38,41)(40,47)(42,51)(43,57)(44,53)(45,61)(48,56)(50,55)(52,59)(54,60)(58,64)(62,63);(1,2,7,12)(3,9,15,24)(4,16,18,22)(5,8,6,11)(10,20,25,17)(13,27,30,31)(14,26,19,21)(23,28,32,29)(33,35)(34,39)(36,43)(37,45)(38,44)(40,52)(41,53)(42,54)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(1,3,7,15)(2,9,12,24)(4,13,18,30)(5,19,6,14)(8,21,11,26)(10,23,25,32)(16,27,22,31)(17,29,20,28)(33,36,38,48)(34,40,41,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(42,56,49,46)(45,62,60,59)(54,64,61,57);(1,4)(2,10)(3,13)(5,17)(6,20)(7,18)(8,22)(9,23)(11,16)(12,25)(14,28)(15,30)(19,29)(21,31)(24,32)(26,27)(34,41)(39,53)(40,55)(42,49)(46,56)(52,63)(54,61)(57,64);(1,5,7,6)(2,11,12,8)(3,14,15,19)(4,17,18,20)(9,21,24,26)(10,16,25,22)(13,28,30,29)(23,31,32,27)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62)Polycyclic presentation:
2^9|b^2=h,c^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273800);