Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251967 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,15)(3,17,21,14)(4,22,27,43)(5,12,9,16)(6,30,33,13)(7,35,19,10)(11,32,39,25)(18,47,50,56)(20,42,54,46)(23,55,28,59)(24,58,40,57)(26,61,48,64)(29,44,34,41)(31,37,51,53)(36,60,63,62)(38,49,45,52);(1,3,8,21)(2,10,15,35)(4,23)(5,19,9,7)(6,31,33,51)(11,24)(12,17,16,14)(13,42,30,46)(18,48)(20,29,54,34)(26,50)(27,28)(36,45)(37,41,53,44)(38,63)(39,40)(47,56)(49,52)(60,62)(61,64);(1,4,9,23)(2,11,16,24)(3,18,7,26)(5,28,8,27)(6,32,34,58)(10,36,14,38)(12,40,15,39)(13,43,44,59)(17,45,35,63)(19,48,21,50)(20,52,31,62)(22,41,55,30)(25,29,57,33)(37,61,42,47)(46,56,53,64)(49,51,60,54);(1,2,9,16)(3,10,7,14)(4,24,23,11)(5,12,8,15)(6,30,34,41)(13,29,44,33)(17,21,35,19)(18,38,26,36)(20,53,31,46)(22,32,55,58)(25,59,57,43)(27,40,28,39)(37,51,42,54)(45,48,63,50)(47,60,61,49)(52,56,62,64)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=i,d^2=i,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fh,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159184);