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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,8,13,25)(4,9,17,27)(5,18,21,11)(7,19,23,14)(12,24,30,26)(15,32,31,22)(16,29,20,28)(33,34,39,42)(35,40,46,54)(36,47,49,58)(37,43,38,51)(41,52,57,48)(44,59,62,64)(45,55,50,56)(53,60,63,61);(1,3)(2,7)(4,15)(5,19)(6,13)(8,18)(9,26)(10,23)(11,25)(12,16)(14,21)(17,31)(20,30)(22,28)(24,27)(29,32)(33,35,39,46)(34,40,42,54)(36,44,49,62)(37,50,38,45)(41,53,57,63)(43,56,51,55)(47,59,58,64)(48,61,52,60);(1,3)(2,8)(4,12)(5,14)(6,13)(7,18)(9,24)(10,25)(11,23)(15,16)(17,30)(19,21)(20,31)(22,28)(26,27)(29,32)(33,36)(34,41)(35,44)(37,48)(38,52)(39,49)(40,53)(42,57)(43,58)(45,60)(46,62)(47,51)(50,61)(54,63)(55,59)(56,64);(1,4,6,17)(2,9,10,27)(3,12,13,30)(5,20,21,16)(7,22,23,32)(8,24,25,26)(11,29,18,28)(14,31,19,15)(34,42)(40,54)(41,57)(43,51)(47,58)(53,63)(55,56)(59,64);(2,10)(3,13)(9,27)(11,18)(12,30)(14,19)(15,31)(28,29)(33,37,39,38)(34,43,42,51)(35,45,46,50)(36,48,49,52)(40,55,54,56)(41,58,57,47)(44,60,62,61)(53,59,63,64)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^2=h,e^2=i,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, 6273893);