Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384777 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2337 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #22395 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,5,12,16,40,8,14,29,41,6,11,28,31)(3,15,20,43,19,9,30,33,21,34,50,58,18,35,49,59)(4,22,26,52,24,39,44,60,27,37,54,56,25,10,36,38)(17,45,48,63,47,42,55,61,23,32,53,57,46,51,62,64);(1,3,16,30,6,18,7,20,8,21,28,49,5,19,29,50)(2,9,31,43,11,15,41,59,14,35,40,58,12,34,13,33)(4,23,44,62,25,47,26,53,27,17,36,55,24,46,54,48)(10,32,56,61,37,42,60,63,39,45,52,64,22,51,38,57);(1,4,5,24,8,27,6,25)(2,10,11,37,14,39,12,22)(3,17,19,47,21,23,18,46)(7,26,16,44,29,54,28,36)(9,32,15,42,35,45,34,51)(13,38,31,56,41,60,40,52)(20,48,30,55,50,53,49,62)(33,57,43,61,59,63,58,64);(3,18,21,19)(7,29)(9,15,35,34)(13,41)(16,28)(17,46,23,47)(20,30,50,49)(26,54)(31,40)(32,42,45,51)(33,58,59,43)(36,44)(38,60)(48,55,53,62)(52,56)(57,64,63,61)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=e,d^2=ef,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,f^a=fi,c^b=ceh,d^b=dh,f^b=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404461);