Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6371551 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#386 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | #62848 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
6003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,8,16,28)(4,17,20,9)(5,10)(6,11)(7,12)(13,25)(14,26,39,52)(15,27,41,54)(18,43,46,29)(19,44,48,30)(21,31)(22,32)(23,33)(24,34)(35,49)(36,50)(37,42)(38,51,40,53)(45,56,47,55)(57,63)(58,61)(59,64)(60,62);(1,3)(2,8,12,28)(4,13)(5,14)(6,15)(7,16)(9,25,17,42)(10,26,32,52)(11,27,34,54)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(29,49,43,61)(30,50,44,62)(31,51,33,53)(45,57)(46,58)(47,59)(48,60)(55,63,56,64);(1,4)(2,9)(3,13)(5,18)(6,19)(7,20)(8,25)(10,29)(11,30)(12,17)(14,35)(15,36)(16,37)(21,45)(22,46)(23,47)(24,48)(26,49)(27,50)(28,42)(31,55)(32,43)(33,56)(34,44)(38,57)(39,58)(40,59)(41,60)(51,63)(52,61)(53,64)(54,62);(1,5,7,22)(2,10,12,32)(3,14,16,39)(4,18,20,46)(6,23,24,21)(8,26,28,52)(9,29,17,43)(11,33,34,31)(13,35,37,58)(15,40,41,38)(19,47,48,45)(25,49,42,61)(27,53,54,51)(30,56,44,55)(36,59,60,57)(50,64,62,63);(1,6,7,24)(2,11,12,34)(3,15,16,41)(4,19,20,48)(5,21,22,23)(8,27,28,54)(9,30,17,44)(10,31,32,33)(13,36,37,60)(14,38,39,40)(18,45,46,47)(25,50,42,62)(26,51,52,53)(29,55,43,56)(35,57,58,59)(49,63,61,64)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=i,e^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,e^d=ei113 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278748);