Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #495899 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21526 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3067 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,15)(4,21,24,39)(5,25,26,38)(6,14,8,12)(9,30)(10,29,36,23)(11,27,37,28)(16,44)(17,41)(18,34)(19,42)(20,43)(22,53,54,35)(31,58)(32,49)(33,50)(40,47)(45,62)(46,59)(48,63)(51,57)(52,56,61,55)(60,64);(1,3,5,18,8,19,28,50,7,20,26,30,6,17,27,49)(2,9,11,15,14,33,25,41,13,34,37,43,12,32,38,42)(4,16,22,46,23,47,55,64,24,48,54,57,29,45,56,58)(10,31,35,40,21,51,52,44,36,60,53,62,39,59,61,63);(1,4)(2,10)(3,16)(5,22)(6,29)(7,24)(8,23)(9,31)(11,35)(12,39)(13,36)(14,21)(15,40)(17,45)(18,46)(19,47)(20,48)(25,52)(26,54)(27,56)(28,55)(30,57)(32,59)(33,51)(34,60)(37,53)(38,61)(41,44)(42,63)(43,62)(49,58)(50,64);(3,17,20,19)(5,26)(9,32,34,33)(11,37)(15,41,43,42)(16,45,48,47)(18,50,30,49)(22,54)(25,38)(27,28)(31,59,60,51)(35,53)(40,44,62,63)(46,64,57,58)(52,61)(55,56)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=dh,g^b=gi,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72285481, #72285570, #72285749, … |
| maximal normal subgroup of | #72285481, #72285570, #72285749, … |
| maximal quotient of | #72287290, #72287495, #72309428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403096);