Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #466646 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16549 |
| 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 | |
874 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)(11,29,41,28)(16,45,48,32)(17,49,53,33)(21,35,47,43)(22,55,57,36)(27,39,56,62)(44,51,58,52)(46,60,64,61)(50,59,54,63);(1,3,6,18,8,19,7,31)(2,9,12,15,14,34,13,42)(4,21,23,48,25,47,24,16)(5,17,29,52,30,53,28,51)(10,35,37,45,38,43,20,32)(11,33,40,44,41,49,26,58)(22,54,56,64,57,50,27,46)(36,61,62,63,55,60,39,59);(1,4,6,24,8,25,7,23)(2,10,12,20,14,38,13,37)(3,16,18,47,19,48,31,21)(5,27,29,57,30,56,28,22)(9,32,15,43,34,45,42,35)(11,39,40,55,41,62,26,36)(17,50,52,64,53,54,51,46)(33,60,44,63,49,61,58,59);(1,5,6,29,8,30,7,28)(2,11,12,40,14,41,13,26)(3,17,18,52,19,53,31,51)(4,22,24,27,25,57,23,56)(9,33,15,44,34,49,42,58)(10,36,20,39,38,55,37,62)(16,46,47,50,48,64,21,54)(32,59,43,60,45,63,35,61)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^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=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373843);