Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432897 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15468 |
| 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 | |
956 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)(3,9,19,33,17,34,18,41)(4,20,23,10,24,38,30,37)(5,25,28,11,29,40,27,39)(15,35,21,48,43,58,44,31)(16,45,46,32,26,55,47,59)(22,36,53,62,54,61,52,49)(42,63,50,64,56,57,51,60);(1,3,6,18,8,17,7,19)(2,9,13,33,12,34,14,41)(4,21,23,15,24,44,30,43)(5,26,27,47,29,16,28,46)(10,35,20,48,37,58,38,31)(11,32,40,45,39,59,25,55)(22,51,52,42,54,50,53,56)(36,60,62,57,61,64,49,63);(1,4)(2,10,12,37)(3,15,17,43)(5,22)(6,23)(7,30)(8,24)(9,31)(11,36,39,61)(13,20,14,38)(16,42,26,56)(18,44,19,21)(25,49,40,62)(27,52)(28,53)(29,54)(32,57)(33,35)(34,48)(41,58)(45,64)(46,50,47,51)(55,60)(59,63);(1,5)(2,11)(3,16)(4,22)(6,28)(7,27)(8,29)(9,32)(10,36)(12,39)(13,25)(14,40)(15,42)(17,26)(18,47)(19,46)(20,49)(21,50)(23,53)(24,54)(30,52)(31,57)(33,55)(34,59)(35,60)(37,61)(38,62)(41,45)(43,56)(44,51)(48,63)(58,64)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340093);