Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #401 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1443 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,13,24,30)(7,15,18,31,17,11,22,26)(8,12,21,29,20,28,23,32)(16,27,33,36,19,25,34,35);(5,7)(6,11)(8,19,20,16)(9,22)(10,17)(12,27,28,25)(13,15)(14,31)(18,24)(21,33,23,34)(26,30)(29,35,32,36);(2,4)(5,8)(6,12,13,28)(7,16)(9,23)(10,20)(11,25,15,27)(14,32,30,29)(17,19)(18,34)(21,24)(22,33)(26,36,31,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,g^b=gi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30982);