Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465718 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16527 |
| 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 | |
832 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)(3,9)(4,19)(5,11,26,38)(6,29)(7,12)(8,13)(10,22)(14,34)(15,44,46,31)(16,49)(17,32)(18,33)(20,50)(21,51,25,55)(23,36)(24,40)(27,57,28,39)(30,42)(35,54,37,53)(41,63,45,62)(43,61)(47,60,48,64)(52,58,56,59);(1,3,7,17,8,18,6,16)(2,9,12,32,13,33,29,49)(4,20,23,43,24,42,22,14)(5,15,28,48,26,46,27,47)(10,34,19,50,36,61,40,30)(11,31,39,60,38,44,57,64)(21,52,54,45,25,56,53,41)(35,62,51,59,37,63,55,58);(1,4,8,24)(2,10,13,36)(3,14,18,43)(5,25,26,21)(6,23,7,22)(9,30,33,50)(11,37,38,35)(12,40,29,19)(15,45,46,41)(16,20,17,42)(27,53,28,54)(31,59,44,58)(32,61,49,34)(39,51,57,55)(47,56,48,52)(60,62,64,63);(1,5,6,27,8,26,7,28)(2,11,12,39,13,38,29,57)(3,15,16,47,18,46,17,48)(4,21,22,53,24,25,23,54)(9,31,32,60,33,44,49,64)(10,35,19,51,36,37,40,55)(14,41,42,56,43,45,20,52)(30,58,34,62,50,59,61,63)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=i,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372915);