Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391960 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14666 |
| 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 | |
850 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)(3,9,17,33)(4,20)(5,11)(6,14)(7,13)(8,12)(10,23)(15,44,21,51)(16,31,49,59)(18,34,19,32)(22,52)(24,37)(25,38)(26,39,50,60)(27,43)(28,41)(29,42)(30,47,35,46)(36,56)(40,54)(45,62,48,64)(53,61,57,58)(55,63);(1,3,7,19,8,17,6,18)(2,9,14,34,12,33,13,32)(4,21,24,46,25,15,23,47)(5,26,27,16,28,50,29,49)(10,35,38,44,37,30,20,51)(11,39,42,59,41,60,43,31)(22,53,54,48,55,57,56,45)(36,62,63,58,40,64,52,61);(1,4,8,25)(2,10,12,37)(3,15,17,21)(5,22,28,55)(6,24,7,23)(9,30,33,35)(11,40,41,36)(13,38,14,20)(16,48,49,45)(18,47,19,46)(26,57,50,53)(27,56,29,54)(31,58,59,61)(32,51,34,44)(39,62,60,64)(42,63,43,52);(1,5)(2,11)(3,16)(4,22)(6,29)(7,27)(8,28)(9,31)(10,36)(12,41)(13,43)(14,42)(15,45)(17,49)(18,26)(19,50)(20,52)(21,48)(23,56)(24,54)(25,55)(30,58)(32,60)(33,59)(34,39)(35,61)(37,40)(38,63)(44,62)(46,57)(47,53)(51,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,f^c=fi,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299159);