Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366710 |
| 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 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,15,20,26)(4,10,5,11)(6,13,7,12)(9,18,27,19)(16,31,21,25)(17,28,32,24)(22,30,23,29)(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);(1,3,8,20)(2,9,14,27)(4,21,5,16)(6,19,7,18)(10,28,11,24)(12,15,13,26)(17,22,32,23)(25,29,31,30)(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);(1,4,8,5)(2,10,14,11)(3,16,20,21)(6,23,7,22)(9,24,27,28)(12,30,13,29)(15,31,26,25)(17,19,32,18)(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);(1,5,8,4)(2,11,14,10)(3,17,20,32)(6,23,7,22)(9,25,27,31)(12,30,13,29)(15,28,26,24)(16,19,21,18)(34,41)(39,53)(40,55)(42,49)(46,56)(52,63)(54,61)(57,64);(1,6,8,7)(2,12,14,13)(3,18,20,19)(4,22,5,23)(9,26,27,15)(10,29,11,30)(16,32,21,17)(24,31,28,25)(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=hi,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,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273977);