Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #22633 |
| prime radical | C2×#41 | C22×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,15,32,48,9,16,31,47,6,13,5,12)(3,17,19,46,20,39,30,49,22,38,33,59,21,10,35,37)(4,23,28,42,27,45,25,50,29,43,34,60,26,11,40,44)(18,51,53,61,24,56,58,63,54,36,52,62,55,41,57,64);(1,3,6,21,9,22,8,20)(2,10,13,38,16,39,15,17)(4,24,27,54,29,55,26,18)(5,30,31,19,32,35,7,33)(11,41,23,51,45,56,43,36)(12,46,47,37,48,59,14,49)(25,53,34,58,40,52,28,57)(42,62,50,64,60,61,44,63);(1,4,6,26,9,29,8,27)(2,11,13,43,16,45,15,23)(3,18,20,24,22,54,21,55)(5,25,31,28,32,40,7,34)(10,36,17,41,39,51,38,56)(12,42,47,44,48,60,14,50)(19,52,30,57,33,53,35,58)(37,61,46,63,49,62,59,64);(1,5,6,31,9,32,8,7)(2,12,13,47,16,48,15,14)(3,19,20,30,22,33,21,35)(4,25,26,28,29,40,27,34)(10,37,17,46,39,49,38,59)(11,42,43,44,45,60,23,50)(18,52,24,57,54,53,55,58)(36,61,41,63,51,62,56,64)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ch,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405001);