Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13873 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1813 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,18,39,19,9,29,33)(4,20,24,10,25,37,23,36)(5,26,27,40,28,11,30,38)(16,42,44,31,21,50,48,34)(17,32,47,59,46,41,45,58)(22,52,53,35,54,49,55,56)(43,62,64,57,51,61,63,60);(1,3,6,18,8,19,7,29)(2,9,13,33,12,15,14,39)(4,21,23,48,25,16,24,44)(5,17,27,45,28,46,30,47)(10,34,20,42,36,31,37,50)(11,32,38,58,26,41,40,59)(22,51,53,64,54,43,55,63)(35,60,49,61,56,57,52,62);(1,4,8,25)(2,10,12,36)(3,16,19,21)(5,22,28,54)(6,24,7,23)(9,31,15,34)(11,35,26,56)(13,37,14,20)(17,43,46,51)(18,48,29,44)(27,55,30,53)(32,57,41,60)(33,42,39,50)(38,52,40,49)(45,64,47,63)(58,61,59,62);(1,5)(2,11)(3,17)(4,22)(6,27)(7,30)(8,28)(9,32)(10,35)(12,26)(13,38)(14,40)(15,41)(16,43)(18,45)(19,46)(20,49)(21,51)(23,53)(24,55)(25,54)(29,47)(31,57)(33,58)(34,60)(36,56)(37,52)(39,59)(42,61)(44,63)(48,64)(50,62)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267487);