Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355552 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,13,25)(4,8,15,22)(5,17,18,9)(7,19,21,14)(12,24,26,32)(16,29,28,23)(20,30,31,27)(33,34)(35,39)(36,46)(37,49)(38,41)(40,47)(42,51)(43,57)(44,53)(45,61)(48,56)(50,55)(52,59)(54,60)(58,64)(62,63);(1,3,5,14,6,13,18,19)(2,7,9,11,10,21,17,25)(4,12,16,27,15,26,28,30)(8,20,23,24,22,31,29,32)(33,35)(34,39)(36,43)(37,45)(38,44)(40,52)(41,53)(42,54)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,24)(13,26)(14,27)(17,29)(18,28)(19,30)(21,31)(25,32)(33,36,38,48)(34,40,41,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(42,56,49,46)(45,62,60,59)(54,64,61,57);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(24,32)(27,30)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62);(3,13)(7,21)(11,25)(12,26)(14,19)(20,31)(24,32)(27,30)(34,41)(39,53)(40,55)(42,49)(46,56)(52,63)(54,61)(57,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72306926, #72307028, #72307036, … |
| maximal normal subgroup of | #72306926, #72307028, #72307036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262758);