Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462062 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16415 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,13,6,12,7,26)(3,9,27,32,16,33,17,39)(4,18,29,49,21,10,22,36)(5,23,28,40,24,11,25,38)(14,30,51,34,42,56,19,48)(15,43,47,58,45,31,46,57)(20,37,54,61,52,35,53,60)(41,55,50,62,44,64,63,59);(1,3,6,16)(2,9)(4,19,21,51)(5,15,24,45)(7,27,8,17)(10,34)(11,31)(12,33)(13,39)(14,29,42,22)(18,48)(20,50,52,63)(23,43)(25,47,28,46)(26,32)(30,36)(35,59)(37,62)(38,58)(40,57)(41,53,44,54)(49,56)(55,61)(60,64);(1,4,6,21)(2,10,12,18)(3,14)(5,20,24,52)(7,22,8,29)(9,30)(11,37,23,35)(13,36,26,49)(15,44)(16,42)(17,19)(25,53,28,54)(27,51)(31,55)(32,34)(33,56)(38,61,40,60)(39,48)(41,45)(43,64)(46,50)(47,63)(57,62)(58,59);(1,5,6,24)(2,11,12,23)(3,15,16,45)(4,20,21,52)(7,28,8,25)(9,31,33,43)(10,35,18,37)(13,40,26,38)(14,41,42,44)(17,47,27,46)(19,50,51,63)(22,54,29,53)(30,55,56,64)(32,58,39,57)(34,59,48,62)(36,61,49,60)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369259);