Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #466414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16544 |
| 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)(3,15)(4,21)(5,11,29,42)(6,13)(7,12)(8,14)(9,18)(10,25)(16,31)(17,48,51,44)(19,34)(20,33)(22,35)(23,55,57,39)(24,38)(26,37)(27,58,56,36)(28,41,30,40)(32,52,61,50)(43,46)(45,59,64,62)(47,53)(49,63,54,60);(1,3,8,20)(2,9)(4,22)(5,17,29,51)(6,18,7,19)(10,35,38,43)(11,32)(12,15)(13,33)(14,34)(16,25)(21,53,37,31)(23,54)(24,47)(26,46)(27,45)(28,50,30,52)(36,60,58,63)(39,62,55,59)(40,48)(41,44)(42,61)(49,57)(56,64);(1,4,7,25,8,26,6,24)(2,10,13,21,14,38,12,37)(3,16,19,46,20,47,18,22)(5,27,30,23,29,56,28,57)(9,31,33,35,34,53,15,43)(11,39,41,36,42,55,40,58)(17,49,52,45,51,54,50,64)(32,60,44,59,61,63,48,62);(1,5,7,28,8,29,6,30)(2,11,13,40,14,42,12,41)(3,17,19,50,20,51,18,52)(4,23,25,27,26,57,24,56)(9,32,33,48,34,61,15,44)(10,36,21,39,38,58,37,55)(16,45,46,49,47,64,22,54)(31,59,35,60,53,62,43,63)Polycyclic presentation:
2^9|a^2=ef,b^2=fg,c^2=ei,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373611);