Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,12,18,34,40,33,17,13,19,11,37,9,16,10,35)(7,20,24,14,32,44,23,51,25,54,63,45,22,43,39,42)(8,26,29,49,27,47,28,15,31,48,36,46,30,52,38,50)(21,55,59,68,56,41,62,66,61,65,58,64,60,67,57,53);(1,2,3,4)(5,7,9,32,13,25,34,22)(6,14,40,42,19,45,16,51)(8,21,27,60,31,61,30,56)(10,24,11,23,33,63,12,39)(15,41,49,65,50,67,46,55)(17,43,37,54,35,44,18,20)(26,64,52,53,48,68,47,66)(28,58,36,62,38,59,29,57);(5,8)(6,15)(7,21)(9,27)(10,36)(11,38)(12,28)(13,31)(14,41)(16,46)(17,48)(18,52)(19,50)(20,53)(22,56)(23,59)(24,62)(25,61)(26,35)(29,33)(30,34)(32,60)(37,47)(39,58)(40,49)(42,65)(43,68)(44,64)(45,67)(51,55)(54,66)(57,63);(5,9,13,34)(6,16,19,40)(7,22,25,32)(8,27,31,30)(10,11,33,12)(14,42,45,51)(15,46,50,49)(17,18,35,37)(20,43,54,44)(21,56,61,60)(23,24,39,63)(26,47,48,52)(28,36,38,29)(41,65,67,55)(53,68,66,64)(57,59,62,58)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72343342, #72343350, #72343353, … |
| maximal normal subgroup of | #72343342, #72343350, #72343353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401763);