Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #435461 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15526 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,13,7,12,6,26)(3,14,17,32,18,9,27,38)(4,19,28,35,21,46,22,10)(5,23,24,37,25,11,29,52)(15,39,42,33,41,57,53,30)(16,36,54,60,44,31,43,58)(20,47,50,51,49,45,55,34)(40,61,64,56,48,63,62,59);(1,3,6,17,7,18,8,27)(2,9,26,38,12,14,13,32)(4,15,28,53,21,41,22,42)(5,16,24,43,25,44,29,54)(10,33,19,39,35,30,46,57)(11,36,52,58,23,31,37,60)(20,48,55,62,49,40,50,64)(34,59,45,61,51,56,47,63);(1,4,8,28,7,21,6,22)(2,10,26,46,12,35,13,19)(3,15,27,53,18,41,17,42)(5,20,29,55,25,49,24,50)(9,30,38,39,14,33,32,57)(11,34,52,47,23,51,37,45)(16,40,54,64,44,48,43,62)(31,56,60,61,36,59,58,63);(1,5)(2,11)(3,16)(4,20)(6,24)(7,25)(8,29)(9,31)(10,34)(12,23)(13,37)(14,36)(15,40)(17,43)(18,44)(19,45)(21,49)(22,50)(26,52)(27,54)(28,55)(30,56)(32,58)(33,59)(35,51)(38,60)(39,61)(41,48)(42,62)(46,47)(53,64)(57,63)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=fi,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342658);