Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391157 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14617 |
| 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 | |
813 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)(3,8,16,30)(4,9)(5,10,22,37)(6,11)(7,24)(12,13)(14,41,43,27)(15,28)(17,38,25,29)(18,31,46,56)(19,48,49,32)(20,39)(21,33)(23,51,40,36)(26,34)(35,44)(42,54)(45,61)(47,58)(50,59,53,62)(52,55)(57,64)(60,63);(1,3,13,17,6,16,7,25)(2,8,24,29,11,30,12,38)(4,18,20,14,21,46,26,43)(5,15,23,52,22,44,40,45)(9,31,34,27,33,56,39,41)(10,35,36,55,37,28,51,61)(19,47,53,63,49,64,50,42)(32,58,62,54,48,57,59,60);(1,4)(2,9,11,33)(3,14)(5,19)(6,21)(7,26)(8,27,30,41)(10,32,37,48)(12,39,24,34)(13,20)(15,42)(16,43)(17,46)(18,25)(22,49)(23,53)(28,54,35,60)(29,56,38,31)(36,62,51,59)(40,50)(44,63)(45,64)(47,52)(55,58,61,57);(1,5,7,23,6,22,13,40)(2,10,24,51,11,37,12,36)(3,15,17,45,16,44,25,52)(4,19,26,53,21,49,20,50)(8,28,38,61,30,35,29,55)(9,32,34,59,33,48,39,62)(14,42,46,64,43,63,18,47)(27,54,31,57,41,60,56,58)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=eh,d^2=efgi,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298354);