Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383753 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,29)(3,14,18,41,19,9,17,33)(4,20,24,36,25,37,23,10)(5,26,27,39,28,40,30,11)(15,34,21,31,45,58,44,42)(16,46,47,32,48,55,49,38)(22,35,52,56,54,61,53,50)(43,62,63,60,64,57,51,59);(1,3,8,19)(2,9,12,14)(4,21,25,44)(5,16,28,48)(6,18,7,17)(10,34,36,58)(11,38,39,32)(13,41,29,33)(15,23,45,24)(20,42,37,31)(22,51,54,63)(26,55,40,46)(27,49,30,47)(35,60,61,59)(43,52,64,53)(50,57,56,62);(1,4)(2,10,12,36)(3,15)(5,22)(6,23)(7,24)(8,25)(9,31,14,42)(11,35,39,61)(13,20,29,37)(16,43)(17,44)(18,21)(19,45)(26,56,40,50)(27,52)(28,54)(30,53)(32,57,38,62)(33,58,41,34)(46,60,55,59)(47,63)(48,64)(49,51);(1,5)(2,11,12,39)(3,16)(4,22)(6,27)(7,30)(8,28)(9,32,14,38)(10,35,36,61)(13,40,29,26)(15,43)(17,47)(18,49)(19,48)(20,50,37,56)(21,51)(23,52)(24,53)(25,54)(31,57,42,62)(33,46,41,55)(34,59,58,60)(44,63)(45,64)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290952);