Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501052 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #32942 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,12,22,25,23,26,28,11,8,9,14,10,15,29,24)(7,13,27,35,18,16,32,36,20,21,17,30,19,31,34,33);(2,4)(5,7)(6,13)(8,21)(9,17)(10,19)(11,20)(12,27)(14,30)(15,31)(16,23)(18,25)(22,35)(24,33)(26,32)(28,36)(29,34);(5,8,11,6)(7,16,20,31)(9,24,12,28)(10,15,25,23)(13,19,21,18)(14,26,22,29)(17,30,27,35)(32,33,34,36);(2,4)Polycyclic presentation:
2^9|a^2=fi,c^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408251);