Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2504 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14137 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1665 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)(3,14)(4,19,23,36)(5,24)(6,28)(7,12)(8,13)(9,17)(10,22,37,30)(11,25)(15,41,20,31)(16,45)(18,33)(21,35,52,59)(26,38)(27,54)(29,34)(32,47)(39,44,48,43)(40,46)(42,62,50,60)(49,56,53,51)(55,58)(57,63,61,64);(1,3,6,17,8,18,7,29)(2,9,12,14,13,34,28,33)(4,20,22,44,23,15,30,43)(5,16,25,46,27,47,26,55)(10,31,36,39,37,41,19,48)(11,32,24,40,38,45,54,58)(21,50,51,64,52,42,56,63)(35,57,53,60,59,61,49,62);(1,4,7,30,8,23,6,22)(2,10,12,36,13,37,28,19)(3,15,17,43,18,20,29,44)(5,21,26,56,27,52,25,51)(9,31,33,48,34,41,14,39)(11,35,24,53,38,59,54,49)(16,42,46,63,47,50,55,64)(32,57,58,62,45,61,40,60);(1,5,6,25,8,27,7,26)(2,11,12,24,13,38,28,54)(3,16,17,46,18,47,29,55)(4,21,22,51,23,52,30,56)(9,32,14,40,34,45,33,58)(10,35,36,53,37,59,19,49)(15,42,43,63,20,50,44,64)(31,57,39,60,41,61,48,62)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281634);