Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #442256 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15706 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,31)(3,14,17,9,19,40,18,34)(4,20,23,10,25,50,24,36)(5,26,29,11,30,38,28,39)(15,42,45,48,21,32,44,41)(16,33,27,58,47,56,46,37)(22,35,53,57,55,62,54,49)(43,61,52,63,51,64,59,60);(1,3,7,17,8,19,6,18)(2,9,12,40,13,34,31,14)(4,21,23,44,25,15,24,45)(5,27,29,47,30,46,28,16)(10,32,50,41,36,42,20,48)(11,37,38,33,39,58,26,56)(22,52,53,51,55,59,54,43)(35,61,57,63,62,64,49,60);(1,4,7,24,8,25,6,23)(2,10,31,50,13,36,12,20)(3,15,18,45,19,21,17,44)(5,22,29,54,30,55,28,53)(9,32,40,48,34,42,14,41)(11,35,26,57,39,62,38,49)(16,43,46,52,47,51,27,59)(33,60,58,64,56,63,37,61);(1,5,7,29,8,30,6,28)(2,11,12,38,13,39,31,26)(3,16,18,46,19,47,17,27)(4,22,24,54,25,55,23,53)(9,33,14,37,34,56,40,58)(10,35,20,49,36,62,50,57)(15,43,45,52,21,51,44,59)(32,60,41,61,42,63,48,64)Polycyclic presentation:
2^9|a^2=eg,b^2=eg,c^2=ei,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 349453);