Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352469 |
| 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 | #18156 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,9)(5,25)(6,16)(10,35)(11,32)(12,42)(13,34)(14,28)(17,50)(21,52)(22,48)(26,55)(29,49)(31,33)(36,61)(37,44)(38,39)(40,59)(41,54)(43,63)(45,57)(46,51)(58,60)(62,64);(1,2,7,14)(3,10,18,37)(4,11,23,41)(5,12,25,31)(6,28,16,9)(8,13,30,45)(15,36,20,38)(17,44,50,35)(19,33,27,42)(21,39,52,61)(22,54,48,32)(24,40,56,64)(26,43,55,58)(29,57,49,34)(46,59,51,62)(47,60,53,63);(1,3,7,18)(2,10,14,37)(4,20,23,15)(5,16,25,6)(8,19,30,27)(9,31,28,12)(11,38,41,36)(13,33,45,42)(17,49,50,29)(21,51,52,46)(22,55,48,26)(24,53,56,47)(32,43,54,58)(34,35,57,44)(39,62,61,59)(40,63,64,60);(1,4,8,24)(2,11,13,40)(3,15,19,47)(5,26,17,21)(6,22,29,51)(7,23,30,56)(9,32,34,59)(10,36,33,60)(12,43,44,39)(14,41,45,64)(16,48,49,46)(18,20,27,53)(25,55,50,52)(28,54,57,62)(31,58,35,61)(37,38,42,63);(1,5,8,17)(2,12,13,44)(3,16,19,49)(4,21,24,26)(6,27,29,18)(7,25,30,50)(9,33,34,10)(11,39,40,43)(14,31,45,35)(15,46,47,48)(20,51,53,22)(23,52,56,55)(28,42,57,37)(32,36,59,60)(38,62,63,54)(41,61,64,58)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259688);