Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349370 |
| 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 | #62831 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,10)(3,12,16,25)(4,9,17,24)(5,11,6,20)(8,14,22,15)(13,26,27,32)(18,23,19,30)(21,28,31,29)(33,34,39,44)(35,45,47,56)(36,50,51,58)(37,43,38,42)(40,49,55,48)(41,52,57,53)(46,60,61,54)(59,62,64,63);(1,3,7,16)(2,8,10,22)(4,13,17,27)(5,15,6,14)(9,21,24,31)(11,25,20,12)(18,29,19,28)(23,32,30,26)(33,35,39,47)(34,40,44,55)(36,46,51,61)(37,49,38,48)(41,54,57,60)(42,56,43,45)(50,64,58,59)(52,63,53,62);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,26)(14,28)(15,29)(16,27)(20,30)(22,31)(25,32)(33,36)(34,41)(35,46)(37,53)(38,52)(39,51)(40,54)(42,50)(43,58)(44,57)(45,59)(47,61)(48,63)(49,62)(55,60)(56,64);(2,10)(4,17)(8,22)(11,20)(12,25)(13,27)(18,19)(28,29)(36,51)(41,57)(46,61)(50,58)(52,53)(54,60)(59,64)(62,63);(1,5,7,6)(2,11,10,20)(3,14,16,15)(4,18,17,19)(8,12,22,25)(9,23,24,30)(13,28,27,29)(21,26,31,32)(35,47)(40,55)(45,56)(46,61)(48,49)(54,60)(59,64)(62,63)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256615);