Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15)(8,11)(9,19)(10,14)(12,17)(13,16)(18,20)(21,22,24,28,25,27,31,34)(23,26,29,33,30,32,35,36);(1,2,3,4)(5,7,16,18,10,17,9,8)(6,11,19,12,14,20,13,15)(21,23,24,29,25,30,31,35)(22,26,28,33,27,32,34,36);(5,8,9,17,10,18,16,7)(6,12,13,11,14,15,19,20);(7,17)(8,18)(11,20)(12,15)(22,27)(26,32)(28,34)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111500);