Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,10,17,11,18,12,16)(7,19,20,13,22,32,21,33)(8,23,24,35,25,34,26,14)(9,27,28,15,30,31,29,36);(5,7,10,20,11,22,12,21)(6,13,16,19,18,33,17,32)(8,9,24,28,25,30,26,29)(14,27,34,36,35,31,23,15);(2,4)(5,8,11,25)(6,14,18,35)(7,9,22,30)(10,26,12,24)(13,31,33,27)(15,19,36,32)(16,23,17,34)(20,29,21,28);(5,9,11,30)(6,15,18,36)(7,8,22,25)(10,29,12,28)(13,23,33,34)(14,19,35,32)(16,31,17,27)(20,26,21,24)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326215);