Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| 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 | C22×C4 | C25 |
| prime radical | #62962 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ 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:
(1,2,10,37)(3,16,20,49)(4,23,38,30)(5,26,12,27)(6,13,36,63)(7,33,32,46)(8,14,9,15)(11,39,42,19)(17,51,53,47)(18,56,40,52)(21,41,22,50)(24,29,25,45)(28,35,31,62)(34,43,61,44)(48,60,64,59)(54,57,55,58);(1,3)(2,11)(4,17)(5,18)(6,19)(7,34)(8,21)(9,22)(10,20)(12,40)(13,41)(14,43)(15,44)(16,33)(23,60)(24,52)(25,56)(26,54)(27,55)(28,57)(29,47)(30,59)(31,58)(32,61)(35,64)(36,39)(37,42)(38,53)(45,51)(46,49)(48,62)(50,63);(1,4,9,27)(2,12,15,30)(3,17,22,55)(5,14,23,37)(6,24,32,62)(7,35,36,25)(8,26,10,38)(11,40,44,59)(13,31,46,45)(16,47,50,57)(18,43,60,42)(19,52,61,48)(20,53,21,54)(28,33,29,63)(34,64,39,56)(41,58,49,51);(1,5,8,30)(2,4,14,26)(3,18,21,59)(6,31,7,29)(9,23,10,12)(11,17,43,54)(13,25,33,62)(15,27,37,38)(16,48,41,56)(19,58,34,47)(20,40,22,60)(24,46,35,63)(28,32,45,36)(39,57,61,51)(42,53,44,55)(49,64,50,52);(1,6,8,7)(2,13,14,33)(3,19,21,34)(4,24,26,35)(5,28,30,45)(9,32,10,36)(11,41,43,16)(12,31,23,29)(15,46,37,63)(17,52,54,64)(18,57,59,51)(20,39,22,61)(25,27,62,38)(40,58,60,47)(42,50,44,49)(48,53,56,55)Polycyclic presentation:
2^9|a^2=hi,c^2=i,d^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260470);