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 | C22×#508 | 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)(5,6,12,19,37,52,36,18,13,20,10,17,9,16,11,39)(7,21,26,45,23,43,25,14,28,44,24,42,27,53,40,46)(8,29,33,51,30,48,38,50,35,54,31,15,34,49,32,47)(22,56,60,55,63,68,59,64,61,65,58,66,57,41,62,67);(1,2,3,4)(5,7,13,28)(6,14,20,46)(8,22,35,61)(9,23,37,27)(10,25,12,40)(11,26,36,24)(15,41,51,68)(16,42,52,45)(17,53,19,43)(18,21,39,44)(29,64,54,67)(30,57,34,63)(31,59,33,62)(32,60,38,58)(47,65,50,56)(48,55,49,66);(5,8)(6,15)(7,22)(9,30)(10,38)(11,33)(12,32)(13,35)(14,41)(16,47)(17,29)(18,48)(19,54)(20,51)(21,55)(23,57)(24,59)(25,58)(26,62)(27,63)(28,61)(31,36)(34,37)(39,49)(40,60)(42,65)(43,67)(44,66)(45,56)(46,68)(50,52)(53,64);(5,9,13,37)(6,16,20,52)(7,23,28,27)(8,30,35,34)(10,36,12,11)(14,42,46,45)(15,47,51,50)(17,18,19,39)(21,43,44,53)(22,57,61,63)(24,40,26,25)(29,48,54,49)(31,32,33,38)(41,65,68,56)(55,67,66,64)(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,f^a=fi,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fhi,g^c=gh,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401713);