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 | #18306 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,22)(3,14,31,50)(4,7,13,8)(5,11,32,49)(6,27,33,21)(10,34,42,17)(12,29,51,30)(15,28,36,47)(16,40,56,63)(18,39,19,38)(20,25,23,26)(24,52,57,48)(35,58,61,44)(37,54,64,55)(41,53,43,62)(45,60,46,59);(1,3,7,19)(2,10,13,36)(4,15,22,42)(5,16,25,46)(6,28,30,17)(8,18,9,31)(11,35,23,43)(12,38,27,14)(20,41,49,61)(21,50,51,39)(24,53,55,44)(26,45,32,56)(29,34,33,47)(37,59,52,40)(48,63,64,60)(54,58,57,62);(6,29)(12,21)(14,39)(17,47)(24,54)(27,51)(28,34)(30,33)(37,48)(38,50)(40,60)(44,62)(52,64)(53,58)(55,57)(59,63);(1,4,8,2)(3,15,18,10)(5,23,26,49)(6,21,29,12)(7,22,9,13)(11,32,20,25)(14,17,39,47)(16,43,45,61)(19,42,31,36)(24,52,54,64)(27,30,51,33)(28,50,34,38)(35,56,41,46)(37,57,48,55)(40,58,60,53)(44,59,62,63);(1,5,8,26)(2,11,4,20)(3,16,18,45)(6,24,29,54)(7,25,9,32)(10,35,15,41)(12,37,21,48)(13,23,22,49)(14,40,39,60)(17,44,47,62)(19,46,31,56)(27,52,51,64)(28,53,34,58)(30,55,33,57)(36,43,42,61)(38,59,50,63)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=i,e^2=i,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, 6260454);