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 |
| prime radical | #63051 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,15,4,18)(5,11,21,26)(6,13,7,12)(9,20,10,17)(16,29,19,31)(22,28,23,27)(24,32,25,30)(33,34,40,44)(35,45,49,56)(36,42,37,51)(38,43,39,53)(41,54,57,48)(46,61,47,60)(50,58,52,59)(55,62,64,63);(1,3,8,4)(2,9,14,10)(5,16,21,19)(6,20,7,17)(11,24,26,25)(12,18,13,15)(22,32,23,30)(27,31,28,29)(33,35,40,49)(34,41,44,57)(36,46,37,47)(38,48,39,54)(42,55,51,64)(43,56,53,45)(50,62,52,63)(58,61,59,60);(1,4,8,3)(2,10,14,9)(5,19,21,16)(6,20,7,17)(11,25,26,24)(12,18,13,15)(22,32,23,30)(27,31,28,29)(33,36,40,37)(34,42,44,51)(35,46,49,47)(38,50,39,52)(41,55,57,64)(43,58,53,59)(45,60,56,61)(48,62,54,63);(1,5,8,21)(2,11,14,26)(3,16,4,19)(6,22,7,23)(9,24,10,25)(12,27,13,28)(15,29,18,31)(17,30,20,32)(33,37,40,36)(34,42,44,51)(35,47,49,46)(38,50,39,52)(41,55,57,64)(43,59,53,58)(45,61,56,60)(48,62,54,63);(5,21)(11,26)(16,19)(22,23)(24,25)(27,28)(29,31)(30,32)(33,38,40,39)(34,43,44,53)(35,48,49,54)(36,50,37,52)(41,56,57,45)(42,58,51,59)(46,62,47,63)(55,61,64,60)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^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, 6259208);