Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510874 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32002 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #28180 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,5,11)(3,15,17,34,18,9,31,36)(4,20,22,44,24,10,32,38)(6,26,29,40,30,13,25,43)(16,45,49,63,48,33,47,61)(19,35,27,41,50,46,60,58)(21,37,55,64,54,52,51,62)(23,39,56,59,57,53,28,42);(1,3)(2,9)(4,21)(5,17)(6,27)(7,31)(8,18)(10,37)(11,36)(12,15)(13,41)(14,34)(16,23)(19,29)(20,52)(22,51)(24,54)(25,50)(26,58)(28,49)(30,60)(32,55)(33,39)(35,43)(38,62)(40,46)(42,61)(44,64)(45,53)(47,56)(48,57)(59,63);(1,4)(2,10)(3,16,18,48)(5,22)(6,28)(7,32)(8,24)(9,33,15,45)(11,38)(12,20)(13,42)(14,44)(17,47,31,49)(19,51,50,55)(21,27,54,60)(23,25)(26,59)(29,57)(30,56)(34,61,36,63)(35,62,46,64)(37,41,52,58)(39,40)(43,53);(1,5,8,7)(2,11,12,14)(3,17,18,31)(4,22,24,32)(6,25,30,29)(9,34,15,36)(10,38,20,44)(13,40,26,43)(16,47,48,49)(19,27,50,60)(21,51,54,55)(23,56,57,28)(33,61,45,63)(35,58,46,41)(37,64,52,62)(39,59,53,42)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=i,g^2=hi,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=gh,h^b=hi,g^c=ghi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72320552, #72320582, #72320677, … |
| maximal normal subgroup of | #72320552, #72320582, #72320677, … |
| maximal quotient of | #72320900, #72321031, #72321087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418071);