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)(4,17)(5,20)(9,27)(10,31)(11,24)(12,21)(14,35)(15,40)(16,42)(19,45)(22,47)(25,30)(26,33)(28,50)(29,55)(32,57)(34,48)(36,38)(37,39)(41,60)(43,53)(49,59)(51,62)(52,54)(56,58)(61,63);(1,2,6,12)(3,9,14,28)(4,10,17,33)(5,21,20,8)(7,11,23,34)(13,30,35,53)(15,29,38,54)(16,43,39,25)(18,31,44,26)(19,32,45,59)(22,48,47,24)(27,37,50,42)(36,56,40,62)(41,58,61,51)(46,57,64,49)(52,60,55,63);(3,14)(5,22)(8,24)(9,28)(13,35)(15,38)(16,42)(18,46)(20,47)(21,48)(25,50)(26,49)(27,43)(29,54)(30,53)(31,57)(36,40)(37,39)(41,63)(44,64)(51,55)(52,58)(56,62)(60,61);(1,3)(2,9)(4,15)(5,16)(6,14)(7,13)(8,25)(10,29)(11,30)(12,28)(17,38)(18,41)(19,36)(20,39)(21,43)(22,37)(23,35)(24,27)(26,51)(31,58)(32,56)(33,54)(34,53)(40,45)(42,47)(44,61)(46,60)(48,50)(49,52)(55,57)(59,62)(63,64);(1,4,7,19)(2,10,11,32)(3,15,13,36)(5,18,22,46)(6,17,23,45)(8,26,24,49)(9,29,30,56)(12,33,34,59)(14,38,35,40)(16,41,37,60)(20,44,47,64)(21,31,48,57)(25,51,27,52)(28,54,53,62)(39,61,42,63)(43,58,50,55)Polycyclic presentation:
2^9|b^2=h,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, 6254428);