Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,18)(14,17)(15,19)(16,20)(21,22,27,29,25,24,26,30)(23,31,34,36,33,28,35,32);(1,2,3,4)(5,7)(6,10)(8,13)(9,14)(11,18)(12,17)(15,20)(16,19)(21,23,25,33)(22,28,24,31)(26,35,27,34)(29,32,30,36);(5,8,9,15)(6,11,12,19)(7,13,14,20)(10,16,17,18)(21,24,25,22)(23,32,33,36)(26,29,27,30)(28,34,31,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fh,e^c=eh,f^c=fh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22960);