Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #389018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14533 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1197 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)(3,13,17,8)(4,18)(5,22,24,36)(6,25)(7,11)(9,21)(10,23,35,27)(12,28)(14,39,41,57)(15,43)(16,31,26,37)(19,32,50,59)(20,48)(29,42,38,53)(30,44)(33,52)(34,46)(40,60)(45,55)(47,54,58,51)(49,56)(61,62)(63,64);(1,3)(2,8)(4,14)(5,15)(6,26)(7,17)(9,29)(10,34)(11,13)(12,37)(16,28)(18,39)(19,49)(20,41)(21,53)(22,55)(23,46)(24,45)(25,31)(27,44)(30,35)(32,56)(33,38)(36,43)(40,50)(42,52)(47,61)(48,57)(51,62)(54,63)(58,64)(59,60);(1,4,6,21,7,20,28,52)(2,9,12,18,11,33,25,48)(3,14,26,53,17,41,16,42)(5,19,27,54,24,50,23,51)(8,29,37,39,13,38,31,57)(10,32,22,47,35,59,36,58)(15,40,44,62,45,49,46,63)(30,56,43,61,34,60,55,64);(1,5)(2,10)(3,15)(4,19)(6,27)(7,24)(8,30)(9,32)(11,35)(12,22)(13,34)(14,40)(16,46)(17,45)(18,47)(20,50)(21,54)(23,28)(25,36)(26,44)(29,56)(31,55)(33,59)(37,43)(38,60)(39,61)(41,49)(42,63)(48,58)(51,52)(53,62)(57,64)Polycyclic presentation:
2^9|a^2=h,c^2=fh,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^d=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296214);