Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512827 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | #13663 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,23)(12,19)(13,20)(14,18)(15,36)(16,37)(17,25)(21,33)(22,34)(24,50)(26,49)(27,60)(28,39)(29,42)(30,48)(31,43)(32,64)(35,45)(38,55)(40,70)(41,54)(44,61)(46,62)(47,63)(51,57)(52,58)(53,68)(56,65)(59,66)(67,72)(69,71);(2,5)(6,8)(9,11,13,25)(10,17,20,23)(12,24,16,28)(14,26,31,55)(15,27,32,56)(18,38,43,49)(19,39,37,50)(21,40,47,51)(22,41,48,52)(29,53,35,59)(30,54,34,58)(33,57,63,70)(36,65,64,60)(42,66,45,68)(44,67,46,69)(61,71,62,72);(10,18)(12,16)(14,33)(15,34)(17,38)(19,45)(20,43)(21,22)(23,49)(24,28)(26,57)(27,58)(29,61)(30,32)(31,63)(35,62)(36,44)(37,42)(39,68)(40,41)(46,64)(47,48)(50,66)(51,52)(53,71)(54,56)(55,70)(59,72)(60,69)(65,67);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,19,20,37)(11,24,25,28)(14,29,31,35)(15,30,32,34)(17,39,23,50)(18,42,43,45)(21,46,47,44)(22,36,48,64)(26,53,55,59)(27,54,56,58)(33,62,63,61)(38,66,49,68)(40,69,51,67)(41,65,52,60)(57,72,70,71)Polycyclic presentation:
2^9|b^2=i,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^c=di,f^c=fg,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72234337, #72234341, #72234348, … |
| maximal normal subgroup of | #72234337, #72234341, #72234348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420201);