Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350061 |
| 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 | #63031 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,11)(3,12,15,21)(4,9,6,19)(5,10,17,25)(8,20,23,13)(14,26,28,31)(16,24,18,29)(22,30,32,27)(33,34)(35,40)(36,47)(37,43)(38,50)(39,42)(41,48)(44,59)(45,55)(46,54)(49,58)(51,56)(52,57)(53,62)(60,64)(61,63);(1,3,7,15)(2,8,11,23)(4,13,6,20)(5,14,17,28)(9,21,19,12)(10,22,25,32)(16,27,18,30)(24,31,29,26)(33,35,39,46)(34,40,42,54)(36,44,49,61)(37,45,38,51)(41,53,57,64)(43,55,50,56)(47,63,58,59)(48,60,52,62);(1,4,7,6)(2,9,11,19)(3,13,15,20)(5,16,17,18)(8,21,23,12)(10,24,25,29)(14,27,28,30)(22,31,32,26)(33,36)(34,41)(35,44)(37,48)(38,52)(39,49)(40,53)(42,57)(43,58)(45,60)(46,61)(47,50)(51,62)(54,64)(55,59)(56,63);(1,5,7,17)(2,10,11,25)(3,14,15,28)(4,16,6,18)(8,22,23,32)(9,24,19,29)(12,26,21,31)(13,27,20,30)(34,42)(40,54)(41,57)(43,50)(47,58)(53,64)(55,56)(59,63);(5,17)(10,25)(14,28)(16,18)(22,32)(24,29)(26,31)(27,30)(33,37,39,38)(34,43,42,50)(35,45,46,51)(36,48,49,52)(40,55,54,56)(41,58,57,47)(44,60,61,62)(53,59,64,63)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257512);