Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #371877 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14082 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1024 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)(3,9,17,33)(4,20)(5,11,25,38)(6,14)(7,13)(8,12)(10,24)(15,34,42,57)(16,43)(18,32,19,31)(21,29,46,48)(22,51,54,49)(23,39)(26,40,28,37)(27,36)(30,45)(35,53,60,52)(41,59)(44,56)(47,58)(50,55)(61,64)(62,63);(1,3)(2,9,12,33)(4,21,23,46)(5,16)(6,18)(7,19)(8,17)(10,34)(11,30,38,43)(13,31,14,32)(15,24,42,27)(20,48)(22,50,54,62)(25,45)(26,44)(28,47)(29,39)(35,59)(36,57)(37,56,40,58)(41,53,61,52)(49,63)(51,55)(60,64);(1,4,6,27,8,23,7,24)(2,10,14,20,12,36,13,39)(3,15,19,21,17,42,18,46)(5,22,28,52,25,54,26,53)(9,29,31,57,33,48,32,34)(11,35,37,49,38,60,40,51)(16,41,44,50,45,61,47,62)(30,55,58,64,43,63,56,59);(1,5,7,26,8,25,6,28)(2,11,14,37,12,38,13,40)(3,16,19,44,17,45,18,47)(4,22,24,53,23,54,27,52)(9,30,32,56,33,43,31,58)(10,35,20,49,36,60,39,51)(15,41,21,50,42,61,46,62)(29,55,34,59,48,63,57,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=ef,d^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279073);