Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351885 |
| 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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8)(3,12)(5,6)(10,21)(11,18)(13,26)(14,19)(16,17)(20,23)(25,28)(27,29)(30,32)(33,34)(35,44)(36,42)(37,40)(38,43)(39,52)(41,47)(45,60)(46,53)(48,56)(49,51)(50,57)(54,58)(55,63)(59,64)(61,62);(1,2,7,11)(3,9,14,24)(4,10,15,25)(5,18,6,8)(12,23,19,20)(13,22,27,31)(16,28,17,21)(26,32,29,30)(33,35,40,48)(34,41,37,46)(36,45,51,64)(38,47,39,53)(42,55,49,61)(43,56,52,44)(50,62,54,63)(57,60,58,59);(1,3)(2,9)(4,13)(5,12)(6,19)(7,14)(8,20)(10,22)(11,24)(15,27)(16,26)(17,29)(18,23)(21,30)(25,31)(28,32)(33,36)(34,42)(35,45)(37,49)(38,50)(39,54)(40,51)(41,55)(43,57)(44,59)(46,61)(47,62)(48,64)(52,58)(53,63)(56,60);(1,4,7,15)(2,10,11,25)(3,13,14,27)(5,16,6,17)(8,21,18,28)(9,22,24,31)(12,26,19,29)(20,30,23,32)(33,37,40,34)(35,46,48,41)(36,49,51,42)(38,43,39,52)(44,47,56,53)(45,61,64,55)(50,57,54,58)(59,62,60,63);(4,15)(10,25)(13,27)(16,17)(21,28)(22,31)(26,29)(30,32)(33,38,40,39)(34,43,37,52)(35,47,48,53)(36,50,51,54)(41,56,46,44)(42,57,49,58)(45,62,64,63)(55,60,61,59)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259203);