Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361285 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2290 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13898 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,13,6,12)(3,15,18,33,20,34,19,9)(4,21,24,38,26,10,25,37)(5,27,28,39,29,40,30,11)(16,42,44,58,45,35,22,31)(17,46,48,41,49,32,47,59)(23,52,54,50,55,56,53,36)(43,57,51,60,62,64,63,61);(1,3)(2,9,13,33)(4,22)(5,17)(6,18)(7,19)(8,20)(10,35,21,42)(11,32,39,46)(12,15,14,34)(16,25)(23,51)(24,45)(26,44)(27,41,40,59)(28,47)(29,49)(30,48)(31,37,58,38)(36,60,50,61)(43,54)(52,57,56,64)(53,62)(55,63);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,36)(12,37)(13,21)(14,38)(15,35)(17,43)(18,22)(19,44)(20,45)(27,56)(28,53)(29,55)(30,54)(32,57)(33,58)(34,42)(39,50)(40,52)(41,60)(46,64)(47,51)(48,63)(49,62)(59,61);(1,5,6,28,8,29,7,30)(2,11,14,40,13,39,12,27)(3,17,18,47,20,49,19,48)(4,23,24,53,26,55,25,54)(9,32,34,59,33,46,15,41)(10,36,38,52,21,50,37,56)(16,43,22,51,45,62,44,63)(31,57,42,61,58,64,35,60)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268626);