Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1441 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,17)(8,13,23,30)(9,16)(10,14)(11,15)(12,19)(18,33,35,32)(20,27)(21,26)(22,31,24,29)(25,34,28,36);(1,2)(3,4)(5,7,10,21,11,20,9,19)(6,12,16,27,15,26,14,17)(8,18,24,36,23,35,22,34)(13,28,31,32,30,25,29,33);(1,3,4,2)(5,8,9,22,11,23,10,24)(6,13,14,29,15,30,16,31)(7,18,19,34,20,35,21,36)(12,25,17,32,26,28,27,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30931);