Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624170 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1498 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)(3,7)(4,6)(5,8)(9,10,16,21)(11,17,23,39)(12,26,29,59)(13,19,15,36)(14,32,35,61)(18,43,46,28)(20,31,48,34)(22,49,50,70)(24,40,30,42)(25,56,54,67)(27,58,37,47)(33,44,62,45)(38,63,64,52)(41,60,66,55)(51,68,53,72)(57,69,71,65);(1,3)(2,5)(4,8)(6,7)(9,11,16,23)(10,17,21,39)(12,22,29,50)(13,30,15,24)(14,25,35,54)(18,38,46,64)(19,42,36,40)(20,41,48,66)(26,49,59,70)(27,53,37,51)(28,52,43,63)(31,60,34,55)(32,56,61,67)(33,57,62,71)(44,69,45,65)(47,68,58,72);(9,12)(10,18)(11,22)(13,27)(14,33)(15,37)(16,29)(17,38)(19,44)(20,47)(21,46)(23,50)(24,51)(25,57)(26,32)(28,34)(30,53)(31,43)(35,62)(36,45)(39,64)(40,65)(41,68)(42,69)(48,58)(49,56)(52,55)(54,71)(59,61)(60,63)(66,72)(67,70);(3,8)(5,7)(11,23)(17,39)(22,50)(24,30)(25,54)(38,64)(40,42)(41,66)(49,70)(51,53)(52,63)(55,60)(56,67)(57,71)(65,69)(68,72);(2,6)(5,7)(9,13,16,15)(10,19,21,36)(11,24,23,30)(12,27,29,37)(14,31,35,34)(17,40,39,42)(18,44,46,45)(20,32,48,61)(22,51,50,53)(25,55,54,60)(26,58,59,47)(28,33,43,62)(38,65,64,69)(41,67,66,56)(49,68,70,72)(52,71,63,57)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531379);