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 | #62933 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ 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:
(1,2,7,13)(3,14,18,28)(4,10,6,25)(5,12,22,37)(8,11,19,32)(9,26,31,15)(16,39,43,54)(17,38,40,53)(20,34,23,47)(21,33,27,50)(24,35,45,36)(29,51,52,42)(30,48,57,41)(44,60,62,61)(46,58,49,59)(55,64,56,63);(1,3)(2,9)(4,15)(5,16)(6,26)(7,18)(8,17)(10,28)(11,29)(12,30)(13,31)(14,25)(19,40)(20,41)(21,42)(22,43)(23,48)(24,44)(27,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,50)(39,47)(45,62)(46,63)(49,64)(58,61)(59,60);(1,4,19,27)(2,10,32,50)(3,15,40,51)(5,20,45,49)(6,8,21,7)(9,28,52,38)(11,33,13,25)(12,34,36,59)(14,29,53,31)(16,41,62,64)(17,42,18,26)(22,23,24,46)(30,54,56,60)(35,58,37,47)(39,55,61,57)(43,48,44,63);(2,11)(5,22)(9,29)(10,33)(12,36)(13,32)(14,38)(16,43)(20,23)(24,45)(25,50)(28,53)(30,56)(31,52)(34,59)(35,37)(39,61)(41,48)(44,62)(46,49)(47,58)(54,60)(55,57)(63,64);(1,5,19,45)(2,12,32,36)(3,16,40,62)(4,20,27,49)(6,23,21,46)(7,22,8,24)(9,30,52,56)(10,34,50,59)(11,35,13,37)(14,39,53,61)(15,41,51,64)(17,44,18,43)(25,47,33,58)(26,48,42,63)(28,54,38,60)(29,55,31,57)Polycyclic presentation:
2^9|a^2=h,c^2=hi,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, 6260015);