Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24658 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13323 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,7,15,31,43,9,13,30,12,8,16,5,29)(3,17,20,44,34,37,19,52,22,10,32,36,21,46,53,61)(4,23,25,40,27,39,60,42,26,57,59,41,28,45,33,11)(18,48,51,38,55,62,58,35,50,63,49,64,24,56,54,47);(1,3)(2,10)(4,24)(5,19)(6,32)(7,34)(8,21)(9,22)(11,38)(12,36)(13,17)(14,44)(15,46)(16,37)(18,27)(20,30)(23,56)(25,58)(26,55)(28,50)(29,61)(31,53)(33,51)(35,40)(39,48)(41,47)(42,64)(43,52)(45,63)(49,60)(54,59)(57,62);(1,4)(2,11)(3,18)(5,25)(6,33)(7,28)(8,27)(9,26)(10,35)(12,39)(13,42)(14,45)(15,41)(16,40)(17,47)(19,49)(20,54)(21,55)(22,50)(23,29)(24,34)(30,60)(31,59)(32,58)(36,62)(37,64)(38,46)(43,57)(44,56)(48,61)(51,53)(52,63);(1,5,8,30,9,31,7,6)(2,12,15,29,13,14,16,43)(3,19,21,20,22,53,34,32)(4,25,27,60,26,59,28,33)(10,36,46,61,17,44,37,52)(11,39,41,23,42,45,40,57)(18,49,55,54,50,51,24,58)(35,62,38,48,47,56,64,63)Polycyclic presentation:
2^9|a^2=f,d^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408726);