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 | |
2611 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,16)(9,20)(14,22)(15,24)(17,23)(19,25)(21,26)(27,29)(28,31)(30,32)(33,34,39,42)(35,45,48,57)(36,41,49,56)(37,44,38,43)(40,51,54,47)(46,60,61,59)(50,58,52,55)(53,62,64,63);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,13)(11,23)(12,24)(18,31)(19,28)(20,29)(21,30)(25,32)(26,27)(33,35)(34,40)(36,46)(37,47)(38,51)(39,48)(41,53)(42,54)(43,45)(44,57)(49,61)(50,62)(52,63)(55,59)(56,64)(58,60);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,26)(11,18)(12,25)(13,27)(15,29)(16,28)(17,30)(23,31)(24,32)(33,36)(34,41)(35,46)(37,50)(38,52)(39,49)(40,53)(42,56)(43,55)(44,58)(45,59)(47,62)(48,61)(51,63)(54,64)(57,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,24,23)(9,18,25,26)(14,28,30,29)(22,31,32,27)(34,42)(36,49)(40,54)(43,44)(45,57)(46,61)(50,52)(62,63);(1,5,7,6)(2,11,12,10)(3,15,17,16)(4,20,21,19)(8,23,24,13)(9,18,25,26)(14,29,30,28)(22,31,32,27)(33,37,39,38)(34,43,42,44)(35,47,48,51)(36,50,49,52)(40,45,54,57)(41,55,56,58)(46,62,61,63)(53,59,64,60)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=h,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, 6259003);