Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563327 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,24)(12,31)(13,20)(14,41)(15,23)(16,22)(17,28)(18,36)(19,40)(21,42)(25,56)(26,45)(27,51)(29,54)(30,46)(32,55)(33,48)(34,64)(35,49)(37,44)(38,52)(39,53)(43,60)(47,59)(50,57)(58,69)(61,72)(62,71)(63,68)(65,67)(66,70);(1,3)(2,5)(4,7)(6,8)(9,11,15,28,16,29,42,30)(10,17,21,24,22,46,23,54)(12,32,35,60,36,25,57,59)(13,37,39,26,40,62,14,27)(18,47,49,56,31,43,50,55)(19,51,53,71,20,45,41,44)(33,63,34,61,38,66,65,58)(48,72,67,68,52,69,64,70);(2,6)(3,7)(9,12)(10,18)(11,25)(13,38)(14,34)(15,35)(16,36)(17,43)(19,52)(20,48)(21,49)(22,31)(23,50)(24,55)(26,61)(27,58)(28,59)(29,32)(30,60)(33,40)(37,63)(39,65)(41,67)(42,57)(44,70)(45,69)(46,47)(51,72)(53,64)(54,56)(62,66)(68,71);(2,6)(5,8)(9,13,15,39,16,40,42,14)(10,19,23,41,22,20,21,53)(11,26,30,37,29,27,28,62)(12,33,35,34,36,38,57,65)(17,44,24,51,46,71,54,45)(18,48,50,64,31,52,49,67)(25,58,60,66,32,61,59,63)(43,68,55,69,47,70,56,72);(9,14,42,40,16,39,15,13)(10,20,23,53,22,19,21,41)(11,27,30,62,29,26,28,37)(12,34,57,33,36,65,35,38)(17,45,54,71,46,51,24,44)(18,48,50,64,31,52,49,67)(25,58,60,66,32,61,59,63)(43,69,56,68,47,72,55,70)Polycyclic presentation:
2^9|b^2=f,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, 7519057);