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 | #63046 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,14)(3,11,15,40)(4,8,20,12)(5,9,25,33)(7,10,16,36)(13,23,30,24)(17,28,41,38)(18,31,42,59)(19,34,43,37)(21,27,44,35)(22,51,49,32)(26,52,29,50)(39,47,57,48)(45,54,60,55)(46,63,61,58)(53,64,56,62);(2,8)(4,20)(5,23)(9,26)(10,27)(11,28)(12,14)(13,32)(17,41)(18,47)(21,44)(22,52)(24,25)(29,33)(30,51)(31,53)(34,54)(35,36)(37,55)(38,40)(39,58)(42,48)(45,60)(46,64)(49,50)(56,59)(57,63)(61,62);(2,9)(4,21)(5,24)(8,26)(10,30)(11,31)(12,29)(13,36)(14,33)(17,45)(18,48)(20,44)(22,49)(23,25)(27,51)(28,53)(32,35)(34,57)(37,39)(38,56)(40,59)(41,60)(42,47)(46,61)(50,52)(54,63)(55,58)(62,64);(2,10)(3,15)(8,27)(9,30)(11,37)(12,35)(13,33)(14,36)(17,41)(18,42)(19,43)(26,51)(28,55)(29,32)(31,39)(34,40)(38,54)(45,60)(46,61)(47,48)(53,58)(56,63)(57,59)(62,64);(1,3,16,43)(2,11,36,37)(4,17,44,60)(5,18,24,48)(6,15,7,19)(8,28,35,55)(9,31,13,39)(10,34,14,40)(12,38,27,54)(20,41,21,45)(22,46,52,64)(23,47,25,42)(26,53,32,58)(29,56,51,63)(30,57,33,59)(49,61,50,62)Polycyclic presentation:
2^9|a^2=h,e^2=hi,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, 6259210);