Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384436 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14379 |
| 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 | |
920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,27)(3,14,17,9,18,33,28,34)(4,10,21,39,22,38,29,37)(5,23,25,11,26,41,30,42)(15,43,46,35,19,50,45,31)(16,47,49,59,48,32,24,40)(20,52,56,62,55,63,54,36)(44,60,57,64,51,58,53,61);(1,3)(2,9)(4,19)(5,24)(6,28)(7,18)(8,17)(10,35)(11,40)(12,33)(13,34)(14,27)(15,22)(16,25)(20,53)(21,45)(23,32)(26,49)(29,46)(30,48)(31,38)(36,61)(37,43)(39,50)(41,47)(42,59)(44,56)(51,54)(52,60)(55,57)(58,63)(62,64);(1,4,8,29,7,22,6,21)(2,10,12,37,13,38,27,39)(3,15,28,46,18,19,17,45)(5,20,30,56,26,55,25,54)(9,31,14,43,34,35,33,50)(11,36,23,52,42,62,41,63)(16,44,49,53,48,51,24,57)(32,58,40,64,47,60,59,61);(1,5,6,25,7,26,8,30)(2,11,12,23,13,42,27,41)(3,16,17,24,18,48,28,49)(4,20,21,54,22,55,29,56)(9,32,14,40,34,47,33,59)(10,36,37,52,38,62,39,63)(15,44,45,57,19,51,46,53)(31,58,43,64,35,60,50,61)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^c=hi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291634);