Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #433835 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15493 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,27)(3,9,16,33,17,34,28,32)(4,18,23,37,21,49,22,10)(5,24,25,38,26,11,29,39)(14,40,19,30,42,35,51,47)(15,43,45,60,46,31,44,59)(20,52,55,57,54,48,56,36)(41,61,53,58,63,62,50,64);(1,3)(2,9)(4,19,21,51)(5,15)(6,28)(7,16)(8,17)(10,35,37,40)(11,31)(12,33)(13,34)(14,23,42,22)(18,47,49,30)(20,53,54,50)(24,43)(25,45)(26,46)(27,32)(29,44)(36,62,57,61)(38,60)(39,59)(41,55,63,56)(48,58,52,64);(1,4,6,22,8,21,7,23)(2,10,12,18,13,37,27,49)(3,14,16,19,17,42,28,51)(5,20,25,55,26,54,29,56)(9,30,32,40,34,47,33,35)(11,36,38,48,24,57,39,52)(15,41,44,50,46,63,45,53)(31,58,59,62,43,64,60,61);(1,5,8,26)(2,11,13,24)(3,15,17,46)(4,20,21,54)(6,25,7,29)(9,31,34,43)(10,36,37,57)(12,38,27,39)(14,41,42,63)(16,44,28,45)(18,48,49,52)(19,50,51,53)(22,55,23,56)(30,58,47,64)(32,59,33,60)(35,61,40,62)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341032);