Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #452545 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16051 |
| 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 | |
869 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)(3,9,17,34)(4,20)(5,25,26,40)(6,14)(7,13)(8,12)(10,24)(11,27,41,28)(15,35,43,60)(16,44)(18,33,19,32)(21,30,48,49)(22,52,55,36)(23,37)(29,38)(31,46)(39,45)(42,62)(47,57)(50,54,58,56)(51,61)(53,64)(59,63);(1,3,6,18,8,17,7,19)(2,9,13,32,12,34,14,33)(4,21,24,43,23,48,29,15)(5,16,27,45,26,47,28,46)(10,35,20,49,38,60,37,30)(11,39,40,44,41,31,25,57)(22,53,54,63,55,51,56,42)(36,61,58,59,52,64,50,62);(1,4,6,29,8,23,7,24)(2,10,14,20,12,38,13,37)(3,15,19,21,17,43,18,48)(5,22,28,54,26,55,27,56)(9,30,32,60,34,49,33,35)(11,36,40,50,41,52,25,58)(16,42,45,51,47,63,46,53)(31,59,44,64,39,62,57,61);(1,5,7,27,8,26,6,28)(2,11,13,25,12,41,14,40)(3,16,18,46,17,47,19,45)(4,22,24,56,23,55,29,54)(9,31,33,57,34,39,32,44)(10,36,37,58,38,52,20,50)(15,42,48,53,43,63,21,51)(30,59,35,61,49,62,60,64)Polycyclic presentation:
2^9|a^2=egh,b^2=f,c^2=ef,d^2=efi,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,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359742);