Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#447 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1300 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)(5,6,12,19,37,52,36,51,13,20,10,17,9,16,11,18)(7,21,26,44,23,42,38,14,28,43,24,41,27,53,25,45)(8,29,33,50,30,47,32,49,35,54,31,15,34,48,39,46)(22,56,60,66,63,68,59,64,61,65,58,55,57,40,62,67);(1,2,3,4)(5,7,13,28)(6,14,20,45)(8,22,35,61)(9,23,37,27)(10,38,12,25)(11,26,36,24)(15,40,50,68)(16,41,52,44)(17,53,19,42)(18,43,51,21)(29,64,54,67)(30,57,34,63)(31,59,33,62)(32,58,39,60)(46,65,49,56)(47,66,48,55);(5,8)(6,15)(7,22)(9,30)(10,39)(11,31)(12,32)(13,35)(14,40)(16,46)(17,54)(18,47)(19,29)(20,50)(21,55)(23,57)(24,62)(25,58)(26,59)(27,63)(28,61)(33,36)(34,37)(38,60)(41,65)(42,64)(43,66)(44,56)(45,68)(48,51)(49,52)(53,67);(5,9,13,37)(6,16,20,52)(7,23,28,27)(8,30,35,34)(10,36,12,11)(14,41,45,44)(15,46,50,49)(17,51,19,18)(21,42,43,53)(22,57,61,63)(24,25,26,38)(29,47,54,48)(31,39,33,32)(40,65,68,56)(55,64,66,67)(58,59,60,62)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fhi,g^c=ghi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401722);