Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347203 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8)(3,13)(4,18)(5,21)(9,27)(10,32)(11,24)(12,22)(14,35)(16,42)(17,44)(20,40)(23,41)(25,31)(26,34)(28,50)(29,48)(30,56)(33,58)(36,38)(37,39)(43,60)(45,53)(49,55)(51,63)(52,54)(57,59)(61,64);(1,2,6,12)(3,9,14,28)(4,10,18,34)(5,22,21,8)(7,11,15,29)(13,31,35,53)(16,30,38,54)(17,45,39,25)(19,32,46,26)(20,33,40,55)(23,48,41,24)(27,37,50,44)(36,57,42,63)(43,59,61,51)(47,58,62,49)(52,60,56,64);(3,14)(5,23)(8,24)(9,28)(13,35)(16,38)(17,44)(19,47)(21,41)(22,48)(25,50)(26,49)(27,45)(30,54)(31,53)(32,58)(36,42)(37,39)(43,64)(46,62)(51,56)(52,59)(57,63)(60,61);(1,3,15,35)(2,9,29,53)(4,16,40,42)(5,17,41,44)(6,14,7,13)(8,25,48,50)(10,30,55,63)(11,31,12,28)(18,38,20,36)(19,43,62,64)(21,39,23,37)(22,45,24,27)(26,51,58,56)(32,59,49,52)(33,57,34,54)(46,61,47,60);(1,4,7,20)(2,10,11,33)(3,16,13,36)(5,19,23,47)(6,18,15,40)(8,26,24,49)(9,30,31,57)(12,34,29,55)(14,38,35,42)(17,43,37,60)(21,46,41,62)(22,32,48,58)(25,51,27,52)(28,54,53,63)(39,61,44,64)(45,59,50,56)Polycyclic presentation:
2^9|b^2=h,d^2=hi,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254442);