Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17167 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,6,11,20,35)(3,13,28,33,16,8,17,40)(4,18,22,37,23,41,29,9)(5,24,27,42,26,10,30,39)(14,44,49,34,19,50,46,31)(15,32,25,38,47,43,48,57)(21,51,54,36,55,63,56,58)(45,64,53,60,52,62,59,61);(1,3,6,16)(2,8,11,13)(4,19)(5,25,26,48)(7,28,20,17)(9,34)(10,38,24,57)(12,40,35,33)(14,23)(15,27,47,30)(18,50)(21,53)(22,46)(29,49)(31,37)(32,39,43,42)(36,62)(41,44)(45,56)(51,60)(52,54)(55,59)(58,64)(61,63);(1,4,20,22,6,23,7,29)(2,9,35,18,11,37,12,41)(3,14,28,46,16,19,17,49)(5,21,30,54,26,55,27,56)(8,31,40,50,13,34,33,44)(10,36,42,63,24,58,39,51)(15,45,25,59,47,52,48,53)(32,60,38,64,43,61,57,62);(1,5,6,26)(2,10,11,24)(3,15,16,47)(4,21,23,55)(7,30,20,27)(8,32,13,43)(9,36,37,58)(12,42,35,39)(14,45,19,52)(17,25,28,48)(18,51,41,63)(22,56,29,54)(31,60,34,61)(33,38,40,57)(44,64,50,62)(46,53,49,59)Polycyclic presentation:
2^9|a^2=efg,b^2=e,c^2=ef,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385976);