Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94694 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7418 |
| commutator | C2×C32 | C2×C4 |
| Frattini, cosocle | C2×M64 | C22 |
| prime radical | #13697 | C2 |
| LCS | G ▷ C2×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C22×C16 ◁ C2×M64 ◁ G | |
| LECS | G ▷ C2×M64 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2)(3,15,18,8)(4,21)(5,26)(6,30)(7,14)(9,16,35,46)(10,23)(11,17)(12,27)(13,28)(19,54,52,42)(20,34,53,44)(22,43)(24,61)(25,40)(29,38)(31,48,32,50)(33,56,45,47)(36,55,63,64)(37,57,62,49)(39,51)(41,60)(58,59);(2,8)(3,16)(4,17)(5,27)(6,28)(9,11)(10,35)(12,42)(13,44)(14,15)(18,46)(19,55)(20,57)(21,32)(22,29)(23,25)(24,60)(26,33)(30,34)(31,40)(36,61)(37,39)(38,45)(41,62)(43,54)(47,50)(48,56)(49,53)(51,59)(52,64)(58,63)Polycyclic presentation:
2^9|a^2=d,c^2=fgi,e^2=ghi,f^2=ghi,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fgi,g^a=gh,h^a=hi,c^b=cf,d^b=defgi,e^b=egi,f^b=fgi,g^b=gh,h^b=hi,d^c=di53 conjugacy classes — view the full character table.
| maximal subgroup of | #72225468 |
| maximal normal subgroup of | #72225468 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1890);