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 | C22×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1103 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)(5,6,11,23,22,33,21,13,10,16,8,14,9,15,25,32)(7,12,24,26,19,28,34,31,20,30,18,27,17,29,35,36);(2,4)(5,7)(6,12)(8,18)(9,17)(10,20)(11,24)(13,31)(14,27)(15,29)(16,30)(19,22)(21,34)(23,26)(25,35)(28,33)(32,36);(6,13)(7,17)(8,21)(9,22)(11,25)(12,26)(14,15)(16,32)(18,24)(19,20)(23,33)(27,30)(28,36)(29,31);(2,4)Polycyclic presentation:
2^9|a^2=fi,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, 408249);