Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9946 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,14,17,11)(8,10)(9,13)(12,18)(15,19,16,20)(21,22)(23,26)(24,31,32,27)(25,28)(29,35,36,33)(30,34);(2,4)(6,10)(7,15)(8,18)(12,13)(16,17)(19,20)(21,23,25,30)(22,26,28,34)(24,29,32,36)(27,35,31,33);(5,7,9,17)(6,11,13,14)(8,16,18,15)(10,19,12,20)(21,24)(22,27)(23,29)(25,32)(26,33)(28,31)(30,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=dh,f^b=fh,d^c=di,f^c=fh,g^c=ghi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22533);