Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351869 |
| 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 | |
2589 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)(4,17)(5,10)(6,12)(7,11)(8,15)(9,18)(14,22)(16,23)(19,25)(20,24)(21,26)(27,30)(28,31)(29,32)(33,34)(35,45)(36,41)(37,44)(38,42)(39,43)(40,48)(46,60)(47,55)(49,54)(50,58)(51,57)(52,56)(53,61)(59,62)(63,64);(1,3)(2,8)(4,14)(5,20)(6,15)(7,16)(9,22)(10,13)(11,24)(12,23)(17,31)(18,30)(19,29)(21,28)(25,27)(26,32)(33,35,39,49)(34,40,43,55)(36,46,51,62)(37,48,38,47)(41,53,57,64)(42,45,44,54)(50,63,52,61)(56,59,58,60);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,22)(10,25)(11,26)(12,17)(13,27)(15,28)(16,29)(20,30)(23,31)(24,32)(33,36)(34,41)(35,46)(37,50)(38,52)(39,51)(40,53)(42,56)(43,57)(44,58)(45,59)(47,61)(48,63)(49,62)(54,60)(55,64);(1,5,7,6)(2,10,11,12)(3,15,16,20)(4,18,19,21)(8,23,24,13)(9,25,26,17)(14,28,29,30)(22,31,32,27)(33,37,39,38)(34,42,43,44)(35,47,49,48)(36,50,51,52)(40,54,55,45)(41,56,57,58)(46,61,62,63)(53,60,64,59);(2,11)(4,19)(8,24)(10,12)(13,23)(14,29)(18,21)(28,30)(34,43)(36,51)(40,55)(42,44)(45,54)(46,62)(50,52)(61,63)Polycyclic presentation:
2^9|b^2=i,d^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,d^b=dh,f^b=fh,g^b=gi,e^c=eh,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, 6259072);