Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473873 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16885 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,12)(3,15,18,9,20,35,19,34)(4,21,24,39,25,38,31,10)(5,26,28,11,30,41,29,42)(16,32,46,43,22,36,47,53)(17,48,50,57,52,33,51,44)(23,37,27,58,55,63,56,40)(45,61,49,60,54,64,59,62);(1,3)(2,9,14,34)(4,22,25,16)(5,17)(6,18)(7,19)(8,20)(10,36)(11,33,42,48)(12,15,13,35)(21,53)(23,54,55,45)(24,47,31,46)(26,57,41,44)(27,59,56,49)(28,50)(29,51)(30,52)(32,39)(37,62)(38,43)(40,64)(58,61)(60,63);(1,4)(2,10)(3,16)(5,27)(6,31)(7,24)(8,25)(9,32)(11,40)(12,21)(13,38)(14,39)(15,43)(17,49)(18,47)(19,46)(20,22)(23,28)(26,37)(29,55)(30,56)(33,61)(34,36)(35,53)(41,63)(42,58)(44,62)(45,50)(48,64)(51,54)(52,59)(57,60);(1,5,8,30)(2,11,14,42)(3,17,20,52)(4,23,25,55)(6,29,7,28)(9,33,34,48)(10,37,39,63)(12,41,13,26)(15,44,35,57)(16,45,22,54)(18,51,19,50)(21,40,38,58)(24,56,31,27)(32,60,36,62)(43,61,53,64)(46,59,47,49)Polycyclic presentation:
2^9|a^2=fg,b^2=egh,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381070);