Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #406110 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14924 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,13,29,11)(3,14,17,8,18,32,48,33)(4,19,23,36,24,39,22,9)(5,25,27,10,28,37,26,38)(15,34,44,49,20,30,43,40)(16,31,46,41,47,59,45,56)(21,52,53,50,55,35,54,57)(42,62,63,60,51,61,64,58);(1,3,7,18)(2,8)(4,20)(5,16,28,47)(6,17,29,48)(9,34,36,30)(10,31)(11,14)(12,32)(13,33)(15,24)(19,49,39,40)(21,51)(22,44)(23,43)(25,56)(26,45,27,46)(35,60,52,58)(37,41)(38,59)(42,55)(50,62,57,61)(53,64)(54,63);(1,4,7,24)(2,9,13,36)(3,15,18,20)(5,21,28,55)(6,22,29,23)(8,30,33,34)(10,35,38,52)(11,19,12,39)(14,40,32,49)(16,42,47,51)(17,43,48,44)(25,57,37,50)(26,53,27,54)(31,58,59,60)(41,61,56,62)(45,63,46,64);(1,5,6,26,7,28,29,27)(2,10,11,37,13,38,12,25)(3,16,17,45,18,47,48,46)(4,21,22,53,24,55,23,54)(8,31,14,41,33,59,32,56)(9,35,19,50,36,52,39,57)(15,42,43,63,20,51,44,64)(30,58,40,61,34,60,49,62)Polycyclic presentation:
2^9|a^2=f,b^2=eh,c^2=i,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313307);