Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2474 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)(5,6,9,14,10,13,25,29)(7,15,18,30,17,11,23,27)(8,19,22,31,21,12,24,28)(16,26,33,35,20,32,34,36);(2,4)(5,7)(6,11)(8,20,21,16)(9,23)(10,17)(12,26,19,32)(13,15)(14,30)(18,25)(22,33,24,34)(27,29)(28,36,31,35);(5,8)(6,12)(7,16)(9,24)(10,21)(11,26)(13,19)(14,31)(15,32)(17,20)(18,34)(22,25)(23,33)(27,36)(28,29)(30,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,f^b=fi,g^b=gi,d^c=di,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, 30771);