Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #429476 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1170 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)(4,20)(5,26)(6,14)(7,12)(8,13)(9,18)(10,24)(11,28)(16,42)(17,45)(19,33)(21,50)(22,34)(23,36)(25,35)(27,38)(29,37)(30,39)(31,49)(32,46)(40,44)(41,47)(43,56)(48,57)(51,53)(52,58)(54,60)(55,59)(61,63)(62,64);(1,3,6,19,8,18,7,30)(2,9,12,33,13,15,14,39)(4,21,24,49,23,16,25,44)(5,17,28,46,27,48,29,47)(10,31,20,50,35,40,36,42)(11,32,26,45,37,41,38,57)(22,43,53,62,54,52,55,63)(34,58,59,64,60,56,51,61);(1,4)(2,10)(3,16)(5,22)(6,25)(7,24)(8,23)(9,31)(11,34)(12,36)(13,35)(14,20)(15,40)(17,43)(18,21)(19,49)(26,51)(27,54)(28,55)(29,53)(30,44)(32,56)(33,42)(37,60)(38,59)(39,50)(41,58)(45,64)(46,63)(47,62)(48,52)(57,61);(1,5,7,28,8,27,6,29)(2,11,14,26,13,37,12,38)(3,17,19,47,18,48,30,46)(4,22,24,55,23,54,25,53)(9,32,33,57,15,41,39,45)(10,34,20,51,35,60,36,59)(16,43,49,62,21,52,44,63)(31,56,42,61,40,58,50,64)Polycyclic presentation:
2^9|b^2=f,d^2=efi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336673);