Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198528 |
| 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 | #60942 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8,11,23)(3,13,16,43)(4,18)(5,22)(10,36,19,28)(12,30,29,54)(14,44,47,41)(15,42,24,17)(20,51)(21,49)(26,40)(27,39)(31,45)(32,33)(37,53)(38,61,50,56)(46,48,52,63)(55,59)(58,62)(60,64);(1,2,5,23,25,29,40,30)(3,9,24,32,42,57,13,39)(4,19,20,28,49,38,51,61)(6,12,22,54,7,11,26,8)(10,37,36,21,50,53,56,18)(14,45,52,55,63,60,41,59)(15,27,17,34,43,33,16,35)(31,46,58,48,64,44,62,47);(1,3)(2,9)(4,14)(5,24)(6,17)(7,16)(8,27)(10,31)(11,35)(12,34)(13,40)(15,26)(18,47)(19,45)(20,52)(21,48)(22,43)(23,32)(25,42)(28,55)(29,57)(30,39)(33,54)(36,58)(37,46)(38,60)(41,51)(44,53)(49,63)(50,64)(56,62)(59,61);(1,4,6,21)(2,10,12,38)(3,14,17,48)(5,20,22,53)(7,18,25,49)(8,28,30,56)(9,31,34,60)(11,19,29,50)(13,41,15,46)(16,47,42,63)(23,36,54,61)(24,52,43,44)(26,37,40,51)(27,55,39,62)(32,58,33,59)(35,45,57,64)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,d^b=di,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, 105726);