Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6451783 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2404 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3751 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,15,19,33)(4,10,23,36)(5,11,28,41)(6,13,7,12)(9,20,32,18)(16,34,46,43)(17,44,48,39)(21,30,47,57)(22,35,52,60)(24,38,25,37)(26,55,49,31)(27,42,29,40)(45,58,51,63)(50,56,64,59)(53,62,54,61);(1,3)(2,9)(4,21)(5,26)(6,20)(7,18)(8,19)(10,34)(11,39)(12,33)(13,15)(14,32)(16,25)(17,27)(22,51)(23,47)(24,46)(28,49)(29,48)(30,38)(31,40)(35,59)(36,43)(37,57)(41,44)(42,55)(45,52)(50,53)(54,64)(56,60)(58,61)(62,63);(1,4)(2,10)(3,16,19,46)(5,22)(6,24)(7,25)(8,23)(9,30,32,57)(11,35)(12,37)(13,38)(14,36)(15,43,33,34)(17,45,48,51)(18,47,20,21)(26,50,49,64)(27,53)(28,52)(29,54)(31,56,55,59)(39,58,44,63)(40,61)(41,60)(42,62);(1,5)(2,11)(3,17,19,48)(4,22)(6,29)(7,27)(8,28)(9,31,32,55)(10,35)(12,42)(13,40)(14,41)(15,44,33,39)(16,45,46,51)(18,49,20,26)(21,50,47,64)(23,52)(24,54)(25,53)(30,56,57,59)(34,58,43,63)(36,60)(37,62)(38,61);(4,23)(10,36)(16,46)(21,47)(22,52)(24,25)(30,57)(34,43)(35,60)(37,38)(45,51)(50,64)(53,54)(56,59)(58,63)(61,62)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^d=fi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358980);