Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384326 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14377 |
| 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 | |
976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,19,9,20,33,18,34)(4,21,26,37,25,36,24,10)(5,11,28,39,29,41,30,40)(16,31,45,58,22,49,44,42)(17,38,27,55,46,59,47,32)(23,48,52,35,53,61,54,62)(43,57,56,60,50,63,51,64);(1,3,8,20)(2,9,14,34)(4,22,25,16)(5,27,29,47)(6,19,7,18)(10,31,37,49)(11,38,41,59)(12,33,13,15)(17,30,46,28)(21,42,36,58)(23,51,53,56)(24,44,26,45)(32,40,55,39)(35,60,62,64)(43,52,50,54)(48,63,61,57);(1,4,6,24,8,25,7,26)(2,10,13,21,14,37,12,36)(3,16,18,44,20,22,19,45)(5,23,28,52,29,53,30,54)(9,31,33,58,34,49,15,42)(11,35,40,48,41,62,39,61)(17,43,27,56,46,50,47,51)(32,57,59,64,55,63,38,60);(1,5,6,28,8,29,7,30)(2,11,12,39,14,41,13,40)(3,17,18,27,20,46,19,47)(4,23,24,52,25,53,26,54)(9,32,15,38,34,55,33,59)(10,35,36,61,37,62,21,48)(16,43,44,56,22,50,45,51)(31,57,42,60,49,63,58,64)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291525);