Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2504 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13924 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,19,33,20,9,18,34)(4,21,25,38,24,37,26,10)(5,27,29,39,30,11,28,40)(16,43,22,31,45,35,49,41)(17,32,47,58,48,42,46,57)(23,50,53,36,52,60,54,55)(44,62,64,61,63,59,51,56);(1,3,6,19,8,20,7,18)(2,9,13,34,12,15,14,33)(4,22,26,45,24,49,25,16)(5,17,29,46,30,48,28,47)(10,35,37,41,38,43,21,31)(11,32,40,57,27,42,39,58)(23,51,53,63,52,64,54,44)(36,59,60,61,55,62,50,56);(1,4,6,25,8,24,7,26)(2,10,14,37,12,38,13,21)(3,16,19,49,20,45,18,22)(5,23,29,54,30,52,28,53)(9,31,33,35,15,41,34,43)(11,36,39,60,27,55,40,50)(17,44,46,64,48,63,47,51)(32,56,58,59,42,61,57,62);(1,5)(2,11)(3,17)(4,23)(6,29)(7,28)(8,30)(9,32)(10,36)(12,27)(13,40)(14,39)(15,42)(16,44)(18,47)(19,46)(20,48)(21,50)(22,51)(24,52)(25,54)(26,53)(31,56)(33,58)(34,57)(35,59)(37,60)(38,55)(41,61)(43,62)(45,63)(49,64)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270067);