Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464970 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16512 |
| 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 | |
1108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,42)(4,20,25,37)(5,26,30,40)(6,13,7,12)(9,31,34,18)(10,23,38,24)(11,28,41,29)(16,45,48,32)(17,33,52,61)(21,35,47,43)(22,36,55,56)(27,57,54,39)(44,51,60,50)(46,62,64,59)(49,63,53,58);(1,3,6,18,8,19,7,31)(2,9,12,15,14,34,13,42)(4,21,23,48,25,47,24,16)(5,17,29,51,30,52,28,50)(10,35,37,45,38,43,20,32)(11,33,26,44,41,61,40,60)(22,53,54,64,55,49,27,46)(36,62,57,63,56,59,39,58);(1,4,6,24,8,25,7,23)(2,10,12,20,14,38,13,37)(3,16,18,47,19,48,31,21)(5,27,29,55,30,54,28,22)(9,32,15,43,34,45,42,35)(11,39,26,56,41,57,40,36)(17,49,51,64,52,53,50,46)(33,59,44,63,61,62,60,58);(1,5)(2,11)(3,17)(4,22)(6,29)(7,28)(8,30)(9,33)(10,36)(12,26)(13,40)(14,41)(15,44)(16,46)(18,51)(19,52)(20,39)(21,53)(23,54)(24,27)(25,55)(31,50)(32,58)(34,61)(35,62)(37,57)(38,56)(42,60)(43,59)(45,63)(47,49)(48,64)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,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, 372167);