Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352701 |
| 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 | #3460 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,18)(3,19,24,52)(4,6,14,34)(5,15,31,13)(7,35,30,46)(8,16,9,17)(11,39,44,21)(12,28,33,27)(20,53,42,63)(22,51,57,54)(23,48,38,41)(25,58,47,32)(26,29,37,45)(36,43,61,49)(40,59,56,62)(50,60,64,55);(1,3)(2,11)(4,20)(5,21)(6,22)(7,36)(8,38)(9,23)(10,24)(12,40)(13,41)(14,42)(15,48)(16,49)(17,43)(18,44)(19,35)(25,51)(26,59)(27,60)(28,55)(29,53)(30,61)(31,39)(32,50)(33,56)(34,57)(37,62)(45,63)(46,52)(47,54)(58,64);(1,4,10,14)(2,12,18,33)(3,20,24,42)(5,25,31,47)(6,17,34,16)(7,37,30,26)(8,27,9,28)(11,40,44,56)(13,45,15,29)(19,50,52,64)(21,51,39,54)(22,43,57,49)(23,55,38,60)(32,46,58,35)(36,62,61,59)(41,63,48,53);(1,5,9,7)(2,13,17,46)(3,21,23,36)(4,25,28,37)(6,32,33,29)(8,30,10,31)(11,41,43,52)(12,45,34,58)(14,47,27,26)(15,16,35,18)(19,44,48,49)(20,51,55,62)(22,50,56,53)(24,39,38,61)(40,63,57,64)(42,54,60,59);(1,6,8,12)(2,14,16,28)(3,22,38,40)(4,17,27,18)(5,29,30,58)(7,32,31,45)(9,33,10,34)(11,42,49,55)(13,26,35,25)(15,37,46,47)(19,51,41,59)(20,43,60,44)(21,53,61,64)(23,56,24,57)(36,50,39,63)(48,62,52,54)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=eh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260056);