Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16893 |
| 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 | |
876 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)(3,14,18,34)(4,20)(5,26)(6,11)(7,13)(8,17,35,19)(9,23)(10,28)(12,31)(15,32,21,36)(16,33,49,62)(22,53)(24,39)(25,38)(27,56)(29,41)(30,42)(37,55)(40,54)(43,51,59,46)(44,48,61,50)(45,64,52,60)(47,58,57,63);(1,3,7,18)(2,8)(4,21)(5,16,29,49)(6,17,31,19)(9,36,39,32)(10,33)(11,14)(12,34)(13,35)(15,25)(20,43,38,59)(22,52)(23,51)(24,46)(26,44)(27,57)(28,48,30,50)(37,63,53,58)(40,64,56,60)(41,61)(42,62)(45,55)(47,54);(1,4,6,23,7,25,31,24)(2,9,12,20,13,39,11,38)(3,15,17,46,18,21,19,51)(5,27,28,55,29,54,30,22)(8,32,34,59,35,36,14,43)(10,40,26,37,42,56,41,53)(16,47,48,52,49,57,50,45)(33,60,44,58,62,64,61,63);(1,5,7,29)(2,10,13,42)(3,16,18,49)(4,22,25,55)(6,30,31,28)(8,33,35,62)(9,37,39,53)(11,26,12,41)(14,44,34,61)(15,45,21,52)(17,50,19,48)(20,40,38,56)(23,54,24,27)(32,58,36,63)(43,64,59,60)(46,57,51,47)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,h^c=hi,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, 381215);