Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479742 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17232 |
| 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 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,15,20,35,18,9,19,34)(4,21,24,38,26,10,25,39)(5,27,29,11,30,43,31,42)(16,44,47,51,46,32,22,36)(17,48,50,33,28,59,49,40)(23,53,55,41,56,58,57,37)(45,64,54,61,60,63,52,62);(1,3,7,20,8,18,6,19)(2,9,13,34,12,15,14,35)(4,22,25,16,26,47,24,46)(5,28,29,50,30,17,31,49)(10,36,38,44,21,51,39,32)(11,40,43,59,42,33,27,48)(23,54,55,45,56,52,57,60)(37,64,53,62,41,63,58,61);(1,4,8,26)(2,10,12,21)(3,16,18,46)(5,23,30,56)(6,25,7,24)(9,32,15,44)(11,41,42,37)(13,39,14,38)(17,45,28,60)(19,47,20,22)(27,58,43,53)(29,57,31,55)(33,62,40,61)(34,51,35,36)(48,64,59,63)(49,54,50,52);(1,5)(2,11)(3,17)(4,23)(6,29)(7,31)(8,30)(9,33)(10,37)(12,42)(13,27)(14,43)(15,40)(16,45)(18,28)(19,49)(20,50)(21,41)(22,52)(24,55)(25,57)(26,56)(32,61)(34,59)(35,48)(36,63)(38,58)(39,53)(44,62)(46,60)(47,54)(51,64)Polycyclic presentation:
2^9|a^2=fg,b^2=eh,c^2=i,g^2=i,h^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=def,f^b=fi,d^c=de,g^c=gi,h^c=hi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386939);