Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30400 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
660 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)(3,9)(4,21)(5,12)(6,10)(7,15)(8,14)(11,40)(13,27)(16,33)(17,48)(18,52)(19,34)(20,35)(22,37)(23,41)(24,44)(25,38)(26,36)(28,42)(29,39)(30,55)(31,47)(32,49)(43,54)(45,57)(46,59)(50,64)(51,63)(53,62)(56,61)(58,60);(1,3,8,16,5,20,7,19)(2,9,15,34,12,35,14,33)(4,17,24,50,22,51,30,46)(6,26,27,54,25,18,29,47)(10,36,39,31,38,52,13,43)(11,32,42,60,41,61,45,62)(21,48,55,59,37,63,44,64)(23,56,28,58,40,49,57,53);(2,10)(3,16)(6,27)(7,8)(9,31)(11,37)(12,38)(13,15)(14,39)(17,46)(18,26)(19,20)(21,41)(23,28)(24,30)(25,29)(32,59)(33,52)(34,36)(35,43)(40,57)(42,55)(44,45)(48,60)(50,51)(53,58)(61,64)(62,63);(1,4)(2,11)(3,17)(5,22)(6,28)(7,24)(8,30)(9,32)(10,37)(12,41)(13,44)(14,42)(15,45)(16,46)(18,53)(19,50)(20,51)(21,38)(23,27)(25,57)(26,58)(29,40)(31,59)(33,60)(34,62)(35,61)(36,63)(39,55)(43,64)(47,49)(48,52)(54,56)Polycyclic presentation:
2^9|b^2=h,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416305);