Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479356 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17201 |
| 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 | |
673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,13,7,26)(3,14,18,32,17,9,27,39)(4,10,28,35,21,36,22,40)(5,11,29,37,24,38,25,41)(15,30,19,50,45,42,48,33)(16,31,47,59,23,43,49,58)(20,34,55,61,53,62,54,63)(44,57,52,64,46,56,51,60);(1,3,6,17)(2,9)(4,19,21,48)(5,23,24,16)(7,27,8,18)(10,33)(11,31)(12,39)(13,14)(15,28,45,22)(20,52,53,51)(25,47,29,49)(26,32)(30,35)(34,60)(36,50)(37,59)(38,43)(40,42)(41,58)(44,55,46,54)(56,63)(57,61)(62,64);(1,4,6,21)(2,10,13,36)(3,15,17,45)(5,20,24,53)(7,28,8,22)(9,30,14,42)(11,34,38,62)(12,40,26,35)(16,46,23,44)(18,48,27,19)(25,55,29,54)(31,57,43,56)(32,50,39,33)(37,63,41,61)(47,52,49,51)(58,64,59,60);(1,5,6,24)(2,11)(3,16,17,23)(4,20,21,53)(7,29,8,25)(9,31)(10,34)(12,41)(13,38)(14,43)(15,44,45,46)(18,49,27,47)(19,51,48,52)(22,55,28,54)(26,37)(30,56)(32,59)(33,60)(35,63)(36,62)(39,58)(40,61)(42,57)(50,64)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386553);