Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1442 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)(7,12)(8,20,22,31)(9,14)(10,16)(11,15)(13,21,29,24)(17,32,34,36)(18,26)(19,27)(23,30)(25,33,28,35);(1,2)(3,4)(5,7,10,18,11,19,9,23)(6,12,16,26,15,27,14,30)(8,17,21,35,22,34,24,33)(13,28,20,32,29,25,31,36);(1,3,4,2)(5,8)(6,13)(7,17)(9,24)(10,21)(11,22)(12,25)(14,31)(15,29)(16,20)(18,35)(19,34)(23,33)(26,36)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fi,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, 30877);