Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624150 |
| 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 |
| prime radical | #61057 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1491 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)(11,17,25,42)(12,26,29,60)(13,19,15,36)(14,32,35,58)(18,43,46,28)(20,31,48,34)(22,49,53,70)(23,39,30,41)(24,55,57,66)(27,59,37,47)(33,44,62,45)(38,63,65,51)(40,61,68,54)(50,67,52,72)(56,69,71,64);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,22)(13,30)(14,24)(15,23)(16,25)(18,38)(19,41)(20,40)(21,42)(26,49)(27,52)(28,51)(29,53)(31,61)(32,55)(33,56)(34,54)(35,57)(36,39)(37,50)(43,63)(44,69)(45,64)(46,65)(47,67)(48,68)(58,66)(59,72)(60,70)(62,71);(9,12,16,29)(10,18,21,46)(11,22,25,53)(13,27,15,37)(14,33,35,62)(17,38,42,65)(19,44,36,45)(20,47,48,59)(23,50,30,52)(24,56,57,71)(26,58,60,32)(28,31,43,34)(39,64,41,69)(40,67,68,72)(49,66,70,55)(51,61,63,54);(3,8)(5,7);(2,6)(5,7)(9,13,16,15)(10,19,21,36)(11,23,25,30)(12,27,29,37)(14,31,35,34)(17,39,42,41)(18,44,46,45)(20,32,48,58)(22,50,53,52)(24,54,57,61)(26,59,60,47)(28,33,43,62)(38,64,65,69)(40,66,68,55)(49,67,70,72)(51,71,63,56)Polycyclic presentation:
2^9|a^2=i,c^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=df,e^b=ei,g^c=gh,h^c=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531359);