Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199552 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,26,46,4,23)(3,17,21,41,53,44,19,34)(5,28,31,47,30,38,27,35)(7,15,24,9,8,16,25,12)(10,36,33,32,13,48,45,29)(11,39,43,18,50,52,49,22)(20,42,56,64,57,37,54,58)(40,63,59,51,62,61,60,55);(2,9,15,14)(4,24)(5,29)(6,25)(10,33,28,47)(11,34,44,43)(12,16,23,46)(13,45,38,35)(17,49,50,41)(19,52)(20,51)(21,39)(27,31)(30,32)(36,48)(37,58,40,59)(42,64,62,60)(54,56)(55,57)(61,63);(2,10)(3,18)(4,25)(5,30)(6,24)(9,33)(11,40)(12,45)(13,46)(14,47)(15,28)(16,38)(17,42)(20,55)(22,53)(23,35)(27,36)(29,32)(31,48)(34,59)(37,44)(41,60)(43,58)(49,64)(50,62)(51,57)(54,56)(61,63);(1,3,8,22)(2,11,16,17)(4,19,24,52)(5,20,32,55)(6,21,25,39)(7,18,26,53)(9,34,23,49)(10,37,38,62)(12,41,14,43)(13,42,28,40)(15,44,46,50)(27,54,36,61)(29,51,30,57)(31,56,48,63)(33,58,35,60)(45,64,47,59)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fhi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106969);