Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351817 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62840 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2603 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,16,23)(4,18,21,25)(5,11,6,10)(8,15,24,17)(9,19,26,20)(14,22,29,32)(27,28,31,30)(33,34,39,43)(35,44,48,54)(36,41,51,58)(37,42,38,52)(40,47,55,46)(45,60,62,59)(49,56,50,57)(53,63,64,61);(1,3,7,16)(2,8,12,24)(4,14,21,29)(5,15,6,17)(9,22,26,32)(10,23,11,13)(18,31,25,27)(19,28,20,30)(33,35,39,48)(34,40,43,55)(36,45,51,62)(37,46,38,47)(41,53,58,64)(42,54,52,44)(49,61,50,63)(56,59,57,60);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,18)(11,25)(12,26)(13,27)(15,30)(16,29)(17,28)(23,31)(24,32)(33,36)(34,41)(35,45)(37,50)(38,49)(39,51)(40,53)(42,57)(43,58)(44,59)(46,63)(47,61)(48,62)(52,56)(54,60)(55,64);(1,5,7,6)(2,10,12,11)(3,15,16,17)(4,19,21,20)(8,23,24,13)(9,25,26,18)(14,28,29,30)(22,27,32,31)(33,37,39,38)(34,42,43,52)(35,46,48,47)(36,49,51,50)(40,54,55,44)(41,56,58,57)(45,61,62,63)(53,59,64,60);(3,16)(8,24)(13,23)(14,29)(15,17)(22,32)(27,31)(28,30)(33,37,39,38)(34,42,43,52)(35,47,48,46)(36,50,51,49)(40,44,55,54)(41,57,58,56)(45,61,62,63)(53,59,64,60)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259049);