Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #494749 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1898 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1937 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)(3,14,19,44)(4,20,23,36)(5,24)(6,26,15,12)(7,11)(8,13)(9,18)(10,21,38,31)(16,45)(17,43,30,42)(22,55)(25,39,29,56)(27,35)(28,53)(32,59,62,64)(33,57)(34,50)(37,54)(40,58)(41,46)(47,52,51,60)(48,61)(49,63);(1,3,15,18,8,19,6,27)(2,9,26,44,13,35,12,14)(4,16,22,51,23,49,28,47)(5,17,29,50,7,30,25,57)(10,32,37,61,38,62,40,41)(11,33,56,43,24,34,39,42)(20,52,53,45,36,60,55,63)(21,46,54,64,31,48,58,59);(1,4)(2,10)(3,16)(5,21)(6,28)(7,31)(8,23)(9,32)(11,36)(12,40)(13,38)(14,41)(15,22)(17,46)(18,51)(19,49)(20,24)(25,58)(26,37)(27,47)(29,54)(30,48)(33,60)(34,52)(35,62)(39,53)(42,45)(43,63)(44,61)(50,64)(55,56)(57,59);(1,5,8,7)(2,11,13,24)(3,17,19,30)(4,21,23,31)(6,25,15,29)(9,33,35,34)(10,36,38,20)(12,39,26,56)(14,42,44,43)(16,46,49,48)(18,50,27,57)(22,54,28,58)(32,60,62,52)(37,55,40,53)(41,45,61,63)(47,59,51,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72303725, #72303735, #72303746, … |
| maximal normal subgroup of | #72303725, #72303735, #72303746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401947);