Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509119 |
| 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)(3,16)(4,22)(5,14)(6,28)(7,12)(8,15)(9,19)(10,25)(11,27)(13,30)(17,47)(18,36)(20,35)(21,37)(23,57)(24,39)(26,56)(29,45)(31,43)(32,44)(33,53)(34,51)(38,61)(40,62)(41,59)(42,54)(46,50)(48,55)(49,63)(52,60)(58,64);(1,3,7,18,8,21,5,19)(2,9,14,37,15,36,12,16)(4,23,24,48,26,17,32,49)(6,20,30,51,31,54,27,52)(10,38,40,64,41,33,45,46)(11,42,43,34,13,35,28,60)(22,55,44,57,56,63,39,47)(25,58,29,61,59,50,62,53);(1,4)(2,10)(3,17)(5,24)(6,29)(7,32)(8,26)(9,33)(11,39)(12,40)(13,44)(14,45)(15,41)(16,46)(18,48)(19,49)(20,53)(21,23)(22,28)(25,30)(27,59)(31,62)(34,55)(35,47)(36,38)(37,64)(42,57)(43,56)(50,51)(52,58)(54,61)(60,63);(2,11)(3,18)(5,7)(6,30)(9,34)(10,39)(12,43)(13,15)(14,28)(16,35)(17,48)(19,21)(20,54)(22,45)(23,49)(24,32)(25,29)(27,31)(33,55)(36,60)(37,42)(38,63)(40,56)(41,44)(46,47)(53,61)(57,64)(59,62)Polycyclic presentation:
2^9|b^2=h,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fh,h^c=hi,e^d=ei,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, 416315);