Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397512 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14751 |
| 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 | |
1195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,18,33,19,41,17,9)(4,20,23,37,24,36,30,10)(5,25,27,56,28,11,26,38)(15,31,43,59,21,34,48,39)(16,44,46,32,47,40,45,54)(22,51,52,35,53,49,57,55)(42,60,62,61,50,58,64,63);(1,3)(2,9,13,33)(4,21,24,15)(5,16)(6,17)(7,18)(8,19)(10,34)(11,32,25,54)(12,14,29,41)(20,39)(22,50,53,42)(23,48,30,43)(26,45)(27,46)(28,47)(31,37)(35,60)(36,59)(38,40,56,44)(49,61)(51,63)(52,64,57,62)(55,58);(1,4,8,24)(2,10,13,37)(3,15,19,21)(5,22,28,53)(6,23,7,30)(9,31,33,34)(11,35,25,55)(12,36,29,20)(14,39,41,59)(16,42,47,50)(17,43,18,48)(26,52,27,57)(32,58,54,60)(38,51,56,49)(40,61,44,63)(45,62,46,64);(1,5,6,26,8,28,7,27)(2,11,12,38,13,25,29,56)(3,16,17,45,19,47,18,46)(4,22,23,52,24,53,30,57)(9,32,14,40,33,54,41,44)(10,35,36,51,37,55,20,49)(15,42,43,62,21,50,48,64)(31,58,39,61,34,60,59,63)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304709);