Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #496611 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21888 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3058 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
741 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15,20,45)(4,21,24,55)(5,12)(6,10)(7,31)(8,14)(9,18,35,16)(11,22,41,25)(13,28)(17,48)(19,32,27,34)(23,42,29,40)(26,38)(30,39)(33,51)(36,47,43,52)(37,57,54,56)(44,46)(49,62)(50,61)(53,60)(58,64)(59,63);(1,3,7,16,8,20,5,18)(2,9,12,15,14,35,31,45)(4,17,25,46,24,51,22,49)(6,27,28,52,30,19,26,47)(10,36,38,34,39,43,13,32)(11,33,21,44,41,48,55,62)(23,53,57,64,29,50,56,63)(37,60,40,59,54,61,42,58);(2,10)(3,16)(5,7)(6,28)(9,32)(11,37)(12,13)(14,39)(15,43)(17,46)(18,20)(19,27)(21,42)(22,25)(23,57)(26,30)(29,56)(31,38)(33,58)(34,35)(36,45)(40,55)(41,54)(44,61)(48,59)(49,51)(50,53)(60,62);(1,4)(2,11,14,41)(3,17)(5,22)(6,29)(7,25)(8,24)(9,33,35,48)(10,37,39,54)(12,21,31,55)(13,42,38,40)(15,44,45,62)(16,46)(18,49)(19,53)(20,51)(23,30)(26,57)(27,50)(28,56)(32,58,34,59)(36,60,43,61)(47,64)(52,63)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=eh,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^b=gi,e^c=ei,f^c=fh,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403807);