Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351787 |
| 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 | #62755 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2612 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)(3,13,17,23)(4,18,19,26)(5,11,6,10)(8,15,24,16)(9,21,25,20)(14,22,28,32)(27,30,31,29)(33,34,38,41)(35,43,46,57)(36,40,47,55)(37,49,52,42)(39,50,54,45)(44,60,61,59)(48,58,51,56)(53,63,64,62);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,23)(11,13)(12,24)(18,31)(19,28)(20,29)(21,30)(25,32)(26,27)(33,35)(34,39)(36,44)(37,50)(38,46)(40,53)(41,54)(42,57)(43,49)(45,52)(47,61)(48,63)(51,62)(55,64)(56,59)(58,60);(1,4)(2,9)(3,14)(5,20)(6,21)(7,19)(8,22)(10,18)(11,26)(12,25)(13,27)(15,29)(16,30)(17,28)(23,31)(24,32)(33,36)(34,40)(35,44)(37,51)(38,47)(39,53)(41,55)(42,58)(43,59)(45,63)(46,61)(48,52)(49,56)(50,62)(54,64)(57,60);(4,19)(9,25)(14,28)(18,26)(20,21)(22,32)(27,31)(29,30)(34,41)(36,47)(39,54)(42,49)(43,57)(44,61)(48,51)(62,63);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,20,19,21)(8,23,24,13)(9,18,25,26)(14,29,28,30)(22,31,32,27)(34,41)(39,54)(40,55)(42,49)(43,57)(53,64)(56,58)(59,60)Polycyclic presentation:
2^9|a^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,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, 6259010);