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 | #3106 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,19,28)(4,10,21,5)(6,12,7,11)(9,17,29,18)(15,16,30,32)(20,23,24,25)(22,31,26,27)(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);(1,3,8,19)(2,9,13,29)(4,15,21,30)(5,22,10,26)(6,18,7,17)(11,14,12,28)(16,25,32,23)(20,27,24,31)(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);(1,4)(2,10)(3,15)(5,13)(6,24)(7,20)(8,21)(9,26)(11,23)(12,25)(14,16)(17,27)(18,31)(19,30)(22,29)(28,32)(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);(1,5,8,10)(2,4,13,21)(3,16,19,32)(6,23,7,25)(9,27,29,31)(11,20,12,24)(14,30,28,15)(17,22,18,26)(34,41)(39,53)(40,55)(42,49)(46,56)(52,63)(54,61)(57,64);(1,6,8,7)(2,11,13,12)(3,17,19,18)(4,20,21,24)(5,23,10,25)(9,28,29,14)(15,31,30,27)(16,22,32,26)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=h,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, 6273805);