Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2622 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14304 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,27)(3,14,19,39,17,9,18,31)(4,20,22,10,23,34,28,33)(5,11,26,35,25,36,24,53)(15,40,43,37,42,49,48,29)(16,38,47,60,45,30,46,57)(21,50,51,32,52,58,54,59)(41,61,62,56,44,63,64,55);(1,3,6,18,8,17,7,19)(2,9,12,31,13,14,27,39)(4,15,22,43,23,42,28,48)(5,16,24,46,25,45,26,47)(10,29,20,49,33,37,34,40)(11,30,35,57,36,38,53,60)(21,41,51,62,52,44,54,64)(32,55,50,63,59,56,58,61);(1,4)(2,10)(3,15,17,42)(5,21)(6,28)(7,22)(8,23)(9,29,14,37)(11,32)(12,34)(13,33)(16,44,45,41)(18,48,19,43)(20,27)(24,54)(25,52)(26,51)(30,56,38,55)(31,40,39,49)(35,58)(36,59)(46,62,47,64)(50,53)(57,63,60,61);(1,5,7,26,8,25,6,24)(2,11,12,35,13,36,27,53)(3,16,19,47,17,45,18,46)(4,21,22,51,23,52,28,54)(9,30,31,57,14,38,39,60)(10,32,34,58,33,59,20,50)(15,41,43,62,42,44,48,64)(29,55,40,61,37,56,49,63)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288847);