Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563327 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,3)(5,6)(7,19)(8,14)(9,15)(10,34)(11,17)(12,18)(13,23)(16,35)(20,48)(21,38)(22,56)(24,39)(25,47)(26,40)(27,58)(28,43)(29,44)(30,49)(31,41)(32,45)(33,46)(36,51)(37,55)(42,59)(50,66)(52,64)(53,68)(54,63)(57,65)(60,67)(61,62);(2,3)(5,7,11,23,12,25,35,24)(6,13,16,19,18,39,17,47)(8,20,28,51,29,53,59,52)(9,30,32,21,33,55,10,22)(14,36,42,48,44,64,43,68)(15,38,34,49,46,56,45,37)(26,57,27,50,31,61,60,54)(40,66,67,65,41,63,58,62);(1,2,4,3)(5,8)(6,14)(7,20)(9,31)(10,27)(11,28)(12,29)(13,36)(15,41)(16,42)(17,43)(18,44)(19,48)(21,54)(22,50)(23,51)(24,52)(25,53)(26,33)(30,61)(32,60)(34,58)(35,59)(37,65)(38,63)(39,64)(40,46)(45,67)(47,68)(49,62)(55,57)(56,66);(1,3,4,2)(5,9,11,32,12,33,35,10)(6,15,17,45,18,46,16,34)(7,21,24,30,25,22,23,55)(8,26,28,27,29,31,59,60)(13,37,19,38,39,49,47,56)(14,40,43,58,44,41,42,67)(20,50,52,57,53,54,51,61)(36,62,48,66,64,65,68,63);(5,10,35,33,12,32,11,9)(6,15,17,45,18,46,16,34)(7,22,24,55,25,21,23,30)(8,27,59,26,29,60,28,31)(13,38,47,37,39,56,19,49)(14,41,43,67,44,40,42,58)(20,50,52,57,53,54,51,61)(36,63,68,65,64,66,48,62)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=h,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521291);