Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383548 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14361 |
| 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 | |
1344 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)(3,14)(4,20,24,10)(5,25,29,40)(6,30)(7,12)(8,13)(9,17)(11,27,41,28)(15,35,45,43)(16,33,47,56)(18,34)(19,42)(21,50,48,32)(22,52)(23,37,31,38)(26,58,46,39)(36,54)(44,63)(49,59)(51,62)(53,60)(55,57)(61,64);(1,3,8,19)(2,9,13,34)(4,21,24,48)(5,26,29,46)(6,18,7,17)(10,35,20,43)(11,39,41,58)(12,42,30,14)(15,31,45,23)(16,28,47,27)(22,53,55,44)(25,56,40,33)(32,37,50,38)(36,60,49,63)(51,59,64,54)(52,61,57,62);(1,4)(2,10)(3,15)(5,22)(6,23)(7,31)(8,24)(9,32)(11,36)(12,37)(13,20)(14,43)(16,44)(17,21)(18,48)(19,45)(25,57)(26,51)(27,54)(28,59)(29,55)(30,38)(33,60)(34,50)(35,42)(39,62)(40,52)(41,49)(46,64)(47,53)(56,63)(58,61);(1,5,6,27,8,29,7,28)(2,11,12,40,13,41,30,25)(3,16,17,26,19,47,18,46)(4,22,23,54,24,55,31,59)(9,33,14,39,34,56,42,58)(10,36,37,52,20,49,38,57)(15,44,21,51,45,53,48,64)(32,60,43,62,50,63,35,61)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^c=hi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290745);