Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510259 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31450 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1764 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,9,28,29)(3,13,27,33,16,8,18,40)(4,19,23,34,21,50,52,10)(5,11,25,39,26,35,43,38)(14,30,48,32,24,41,17,42)(15,45,55,31,20,51,47,36)(22,49,57,60,56,63,64,37)(44,59,54,61,53,62,46,58);(1,3,6,27,7,16,28,18)(2,8,29,33,9,13,12,40)(4,20,52,55,21,15,23,47)(5,24,25,17,26,14,43,48)(10,36,19,45,34,31,50,51)(11,30,38,42,35,41,39,32)(22,44,64,46,56,53,57,54)(37,61,49,62,60,58,63,59);(2,9)(3,14,16,24)(5,25)(6,28)(8,30,13,41)(10,34)(11,38)(15,44,20,53)(17,18,48,27)(22,57)(23,52)(26,43)(31,58,36,61)(32,40,42,33)(35,39)(37,63)(45,59,51,62)(46,47,54,55)(49,60)(56,64);(1,4)(2,10)(3,15)(5,22)(6,23)(7,21)(8,31)(9,34)(11,37)(12,19)(13,36)(14,44)(16,20)(17,46)(18,47)(24,53)(25,57)(26,56)(27,55)(28,52)(29,50)(30,58)(32,59)(33,45)(35,60)(38,63)(39,49)(40,51)(41,61)(42,62)(43,64)(48,54)Polycyclic presentation:
2^9|a^2=eh,b^2=efh,c^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,g^c=gh,h^c=hi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417455);