Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60959 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8,11,22)(3,13,16,42)(4,18)(5,21)(10,35,36,27)(12,29,28,52)(14,43,45,40)(15,41,23,17)(19,49)(20,48)(25,39)(26,38)(30,57)(31,32)(37,61,54,55)(44,46,50,63)(47,51)(53,59)(58,62)(60,64);(1,2,5,22,24,28,39,29)(3,9,23,31,41,56,13,38)(4,10,19,35,48,54,49,55)(6,12,21,52,7,11,25,8)(14,30,50,58,63,64,40,62)(15,26,17,33,42,32,16,34)(18,36,47,27,20,37,51,61)(43,59,45,57,44,53,46,60);(1,3)(2,9)(4,14)(5,23)(6,17)(7,16)(8,26)(10,30)(11,34)(12,33)(13,39)(15,25)(18,45)(19,50)(20,46)(21,42)(22,31)(24,41)(27,53)(28,56)(29,38)(32,52)(35,58)(36,57)(37,60)(40,49)(43,51)(44,47)(48,63)(54,64)(55,62)(59,61);(1,4,6,20)(2,10,12,37)(3,14,17,46)(5,19,21,51)(7,18,24,48)(8,27,29,55)(9,30,33,60)(11,36,28,54)(13,40,15,44)(16,45,41,63)(22,35,52,61)(23,50,42,43)(25,47,39,49)(26,53,38,62)(31,58,32,59)(34,57,56,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105702);