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 | #62908 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,36)(3,17,37,62)(4,12,13,29)(5,26,38,25)(6,14,35,60)(7,32,31,46)(8,15,9,16)(11,39,61,20)(18,50,41,64)(19,51,63,59)(21,49,22,42)(23,45,24,28)(27,58,30,34)(33,44,57,43)(40,48,56,52)(47,54,55,53);(1,3)(2,11)(4,18)(5,19)(6,20)(7,33)(8,22)(9,21)(10,37)(12,40)(13,41)(14,42)(15,44)(16,43)(17,32)(23,51)(24,59)(25,54)(26,53)(27,47)(28,50)(29,56)(30,55)(31,57)(34,48)(35,39)(36,61)(38,63)(45,64)(46,62)(49,60)(52,58);(1,4,10,13)(2,12,36,29)(3,18,37,41)(5,16,38,15)(6,23,35,24)(7,34,31,58)(8,25,9,26)(11,40,61,56)(14,45,60,28)(17,47,62,55)(19,43,63,44)(20,51,39,59)(21,53,22,54)(27,46,30,32)(33,48,57,52)(42,64,49,50);(1,5,8,12)(2,13,15,26)(3,19,22,40)(4,16,25,36)(6,30,7,28)(9,29,10,38)(11,41,44,53)(14,23,32,34)(17,48,42,51)(18,43,54,61)(20,55,33,50)(21,56,37,63)(24,46,58,60)(27,31,45,35)(39,47,57,64)(49,59,62,52);(1,6,8,7)(2,14,15,32)(3,20,22,33)(4,23,25,34)(5,27,12,45)(9,31,10,35)(11,42,44,17)(13,24,26,58)(16,46,36,60)(18,51,54,48)(19,47,40,64)(21,57,37,39)(28,38,30,29)(41,59,53,52)(43,62,61,49)(50,63,55,56)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,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, 6260490);