Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476705 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17095 |
| 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 | |
982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,31)(3,14,18,34,19,43,17,9)(4,20,24,36,25,40,23,10)(5,26,29,57,30,11,28,39)(15,32,46,51,21,52,45,41)(16,42,49,37,50,64,48,33)(22,38,54,62,55,56,27,35)(44,61,58,63,53,59,47,60);(1,3,6,17,8,19,7,18)(2,9,12,14,13,34,31,43)(4,21,23,46,25,15,24,45)(5,16,28,48,30,50,29,49)(10,32,20,51,36,52,40,41)(11,37,39,64,26,33,57,42)(22,53,27,58,55,44,54,47)(35,61,38,63,62,59,56,60);(1,4)(2,10)(3,15)(5,27)(6,24)(7,23)(8,25)(9,32)(11,38)(12,40)(13,36)(14,41)(16,47)(17,46)(18,45)(19,21)(20,31)(22,28)(26,56)(29,55)(30,54)(33,60)(34,52)(35,39)(37,63)(42,59)(43,51)(44,48)(49,53)(50,58)(57,62)(61,64);(1,5,6,28,8,30,7,29)(2,11,12,39,13,26,31,57)(3,16,17,48,19,50,18,49)(4,22,23,27,25,55,24,54)(9,33,14,42,34,37,43,64)(10,35,20,38,36,62,40,56)(15,44,45,47,21,53,46,58)(32,59,51,60,52,61,41,63)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383901);