Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #398812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14769 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
815 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)(3,9,27,32,16,33,17,37)(4,18,28,35,21,46,22,10)(5,23,29,53,24,11,25,36)(14,38,48,57,19,44,40,30)(15,41,54,59,42,31,43,58)(20,49,55,34,50,45,51,52)(39,61,64,56,47,60,62,63);(1,3,6,16)(2,9,13,33)(4,19,21,14)(5,15,24,42)(7,27,8,17)(10,30,35,57)(11,31,23,41)(12,37,26,32)(18,44,46,38)(20,47,50,39)(22,40,28,48)(25,54,29,43)(34,56,52,63)(36,59,53,58)(45,61,49,60)(51,62,55,64);(1,4,7,22,6,21,8,28)(2,10,12,18,13,35,26,46)(3,14,17,40,16,19,27,48)(5,20,25,51,24,50,29,55)(9,30,32,38,33,57,37,44)(11,34,36,45,23,52,53,49)(15,39,43,62,42,47,54,64)(31,56,58,60,41,63,59,61);(1,5,6,24)(2,11)(3,15,16,42)(4,20,21,50)(7,25,8,29)(9,31)(10,34)(12,36)(13,23)(14,39,19,47)(17,43,27,54)(18,45)(22,51,28,55)(26,53)(30,56)(32,58)(33,41)(35,52)(37,59)(38,60)(40,62,48,64)(44,61)(46,49)(57,63)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 306009);