Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349684 |
| 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 | #62827 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2626 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)(4,9)(5,12)(6,10)(7,11)(8,15)(14,27)(16,26)(17,25)(18,24)(19,23)(20,22)(21,30)(28,32)(29,31)(33,34,39,42)(35,45,47,55)(36,50,51,58)(37,44,38,43)(40,49,56,48)(41,52,57,53)(46,60,61,54)(59,62,64,63);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,13)(11,22)(12,26)(17,28)(18,29)(19,30)(23,31)(24,32)(25,27)(33,35)(34,40)(36,46)(37,49)(38,48)(39,47)(41,54)(42,56)(43,55)(44,45)(50,64)(51,61)(52,63)(53,62)(57,60)(58,59);(1,4)(2,9)(3,14)(5,19)(6,17)(7,18)(8,21)(10,25)(11,24)(12,23)(13,27)(15,30)(16,29)(20,28)(22,32)(26,31)(33,36)(34,41)(35,46)(37,53)(38,52)(39,51)(40,54)(42,57)(43,50)(44,58)(45,59)(47,61)(48,63)(49,62)(55,64)(56,60);(1,5,7,6)(2,10,11,12)(3,15,16,20)(4,17,18,19)(8,13,22,26)(9,23,24,25)(14,28,29,30)(21,31,32,27)(36,51)(41,57)(46,61)(50,58)(52,53)(54,60)(59,64)(62,63);(2,11)(3,16)(9,24)(10,12)(14,29)(15,20)(23,25)(28,30)(34,42)(35,47)(41,57)(43,44)(46,61)(48,49)(50,58)(62,63)Polycyclic presentation:
2^9|a^2=i,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256917);