Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30593 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33618 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
525 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)(3,8)(4,18)(5,11)(6,24)(7,28)(9,20)(10,22)(12,27)(13,23)(14,43)(15,31)(16,34)(17,51)(19,36)(21,55)(25,57)(26,42)(29,41)(30,44)(32,49)(33,50)(35,54)(37,53)(38,56)(39,47)(40,59)(45,64)(46,63)(48,62)(52,61)(58,60);(1,3,7,17,5,16,23,50)(2,8,13,33,11,34,28,51)(4,14,21,45,19,46,29,52)(6,25,27,59,22,15,47,49)(9,35,38,62,37,30,42,60)(10,31,12,40,24,57,39,32)(18,43,41,61,36,63,55,64)(20,54,26,58,53,44,56,48);(1,4)(2,9)(3,14,16,46)(5,19)(6,26)(7,29)(8,30,34,35)(10,36)(11,37)(12,41)(13,42)(15,48,25,58)(17,52,50,45)(18,24)(20,27)(21,23)(22,56)(28,38)(31,43,57,63)(32,61,40,64)(33,62,51,60)(39,55)(44,59,54,49)(47,53);(2,10)(6,27)(7,23)(8,31)(9,36)(11,24)(12,28)(13,39)(15,49)(17,50)(18,37)(20,26)(21,29)(22,47)(25,59)(30,43)(32,33)(34,57)(35,63)(38,41)(40,51)(42,55)(44,48)(45,52)(53,56)(54,58)(60,64)(61,62)Polycyclic presentation:
2^9|b^2=h,c^2=ei,f^2=hi,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,g^b=gi,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, 416542);