Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500164 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24033 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61051 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,9,18,41)(4,19,21,44)(5,25,6,31)(7,35,29,13)(10,23,43,22)(11,34,12,30)(15,42,53,58)(16,49,28,46)(17,51,36,40)(20,38,59,57)(24,47,37,50)(26,48,32,39)(27,45,33,52)(54,62,56,64)(55,61,60,63);(1,3)(2,9)(4,20)(5,26)(6,32)(7,36)(8,18)(10,42)(11,46)(12,49)(13,51)(14,41)(15,23)(16,34)(17,29)(19,57)(21,59)(22,53)(24,55)(25,48)(27,56)(28,30)(31,39)(33,54)(35,40)(37,60)(38,44)(43,58)(45,62)(47,63)(50,61)(52,64);(1,4,7,37,8,21,29,24)(2,10,13,52,14,43,35,45)(3,15,17,56,18,53,36,54)(5,27,30,23,6,33,34,22)(9,38,40,63,41,57,51,61)(11,47,25,44,12,50,31,19)(16,55,32,59,28,60,26,20)(39,62,49,58,48,64,46,42);(1,3)(2,9)(4,15)(5,28)(6,16)(7,17)(8,18)(10,38)(11,48)(12,39)(13,40)(14,41)(19,58)(20,23)(21,53)(22,59)(24,54)(25,46)(26,30)(27,60)(29,36)(31,49)(32,34)(33,55)(35,51)(37,56)(42,44)(43,57)(45,61)(47,64)(50,62)(52,63)Polycyclic presentation:
2^9|a^2=i,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=gh,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,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, #72249766, #72249831, … |
| maximal normal subgroup of | #72249754, #72249766, #72249831, … |
| maximal quotient of | #72249944, #72250087, #72250137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407362);