Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #409366 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15018 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
760 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)(3,13,16,8)(4,9)(5,22)(6,25)(7,11)(10,24)(12,27)(14,31,18,28)(15,41,42,50)(17,30,44,38)(19,47)(20,36)(21,33)(23,52)(26,34)(29,43,37,60)(32,49)(35,53)(39,61,46,63)(40,56,45,57)(48,51)(54,59)(55,62,58,64);(1,3,6,17,7,16,27,44)(2,8,25,38,11,13,12,30)(4,18,20,45,21,14,26,40)(5,15,24,43,23,42,53,60)(9,31,36,56,33,28,34,57)(10,29,52,41,35,37,22,50)(19,46,49,64,48,39,54,62)(32,58,51,63,59,55,47,61);(1,4,7,21)(2,9,11,33)(3,14,16,18)(5,19,23,48)(6,26,27,20)(8,28,13,31)(10,32,35,59)(12,36,25,34)(15,39,42,46)(17,45,44,40)(22,51,52,47)(24,54,53,49)(29,55,37,58)(30,57,38,56)(41,63,50,61)(43,64,60,62);(1,5)(2,10,11,35)(3,15)(4,19)(6,24)(7,23)(8,29,13,37)(9,32,33,59)(12,22,25,52)(14,39)(16,42)(17,43)(18,46)(20,49)(21,48)(26,54)(27,53)(28,55,31,58)(30,50,38,41)(34,47,36,51)(40,62)(44,60)(45,64)(56,63,57,61)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 316563);