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 | #18290 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2609 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)(3,17,21,41)(4,24,38,6)(5,14,36,46)(7,33,30,13)(8,15,9,16)(11,39,42,19)(12,28,32,27)(18,51,53,48)(20,49,58,63)(22,47,23,50)(25,29,26,45)(31,61,60,35)(34,43,56,44)(40,52,59,62)(54,64,55,57);(1,3,8,22)(2,11,15,43)(4,18,27,54)(5,19,7,34)(6,20,32,59)(9,23,10,21)(12,40,24,58)(13,41,46,50)(14,47,33,17)(16,44,37,42)(25,52,35,63)(26,62,61,49)(28,55,38,53)(29,48,31,57)(30,56,36,39)(45,51,60,64);(1,4)(2,12)(3,18)(5,25)(6,16)(7,35)(8,27)(9,28)(10,38)(11,40)(13,45)(14,31)(15,24)(17,48)(19,52)(20,44)(21,53)(22,54)(23,55)(26,36)(29,33)(30,61)(32,37)(34,63)(39,62)(41,51)(42,59)(43,58)(46,60)(47,57)(49,56)(50,64);(1,5,8,7)(2,13,15,46)(3,19,22,34)(4,25,27,35)(6,31,32,29)(9,30,10,36)(11,41,43,50)(12,45,24,60)(14,37,33,16)(17,44,47,42)(18,52,54,63)(20,57,59,48)(21,39,23,56)(26,28,61,38)(40,51,58,64)(49,53,62,55);(1,6,9,12)(2,4,16,28)(3,20,23,40)(5,29,30,60)(7,31,36,45)(8,32,10,24)(11,18,44,55)(13,35,14,26)(15,27,37,38)(17,49,50,52)(19,48,56,64)(21,58,22,59)(25,33,61,46)(34,57,39,51)(41,63,47,62)(42,53,43,54)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=h,e^2=i,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, 6260094);