Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349298 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2605 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,14)(13,27)(15,23)(16,22)(17,24)(18,25)(19,26)(21,29)(28,31)(30,32)(33,34)(35,45)(36,50)(37,44)(38,43)(39,42)(40,48)(41,53)(46,60)(47,56)(49,55)(51,57)(52,58)(54,61)(59,63)(62,64);(1,3,7,16)(2,8,11,23)(4,13,19,30)(5,14,6,15)(9,21,26,31)(10,22,20,12)(17,28,18,29)(24,27,25,32)(33,35,39,47)(34,40,42,55)(36,46,51,61)(37,49,38,48)(41,54,58,60)(43,56,44,45)(50,64,57,59)(52,63,53,62);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,21)(10,25)(11,26)(12,27)(14,29)(15,28)(16,30)(20,24)(22,32)(23,31)(33,36)(34,41)(35,46)(37,53)(38,52)(39,51)(40,54)(42,58)(43,50)(44,57)(45,59)(47,61)(48,63)(49,62)(55,60)(56,64);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,17,19,18)(8,22,23,12)(9,24,26,25)(13,28,30,29)(21,27,31,32)(34,42)(36,51)(40,55)(43,44)(45,56)(46,61)(52,53)(62,63);(1,5,7,6)(2,10,11,20)(3,15,16,14)(4,18,19,17)(8,12,23,22)(9,25,26,24)(13,28,30,29)(21,27,31,32)(35,47)(40,55)(45,56)(46,61)(48,49)(54,60)(59,64)(62,63)Polycyclic presentation:
2^9|b^2=h,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256565);