Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500351 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24089 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61260 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
709 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)(5,10)(6,27)(7,31)(8,16)(9,23)(11,30)(12,44)(13,22)(15,51)(17,45)(18,41)(19,36)(21,34)(24,42)(25,38)(26,47)(28,43)(29,39)(32,48)(33,46)(35,57)(37,54)(40,60)(49,59)(50,52)(53,64)(55,63)(56,61)(58,62);(1,3)(2,8)(4,21,25,52)(5,16)(6,28)(7,32)(9,37,40,51)(10,14)(11,41)(12,45)(13,48)(15,23,54,60)(17,30)(18,44)(19,22)(20,34,38,50)(24,56,33,55)(26,57)(27,35)(29,58,59,53)(31,36)(39,62,49,64)(42,61,46,63)(43,47);(1,4,22,24,5,25,7,33)(2,9,31,39,10,40,13,49)(3,15,32,53,16,54,19,58)(6,29,44,23,26,59,30,60)(8,34,48,61,14,50,36,63)(11,42,47,20,12,46,27,38)(17,55,57,52,18,56,28,21)(35,62,45,37,43,64,41,51);(1,3)(2,8)(4,15)(5,16)(6,18)(7,19)(9,34)(10,14)(11,43)(12,35)(13,36)(17,26)(20,51)(21,23)(22,32)(24,53)(25,54)(27,45)(28,44)(29,56)(30,57)(31,48)(33,58)(37,38)(39,61)(40,50)(41,47)(42,64)(46,62)(49,63)(52,60)(55,59)Polycyclic presentation:
2^9|b^2=ei,c^2=eh,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,e^c=ei,f^c=fhi,g^c=ghi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72249754, #72249759, #72249825, … |
| maximal normal subgroup of | #72249754, #72249759, #72249825, … |
| maximal quotient of | #72240831, #72249927, #72250069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407548);