Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ 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)(7,11)(8,18,19,12)(9,14)(10,13)(15,22,28,25)(16,23)(17,24)(20,31,32,26)(21,27)(29,33,36,35)(30,34);(1,2)(3,4)(5,7,9,17,10,16,21,30)(6,11,14,24,13,23,27,34)(8,15,20,29,19,28,32,36)(12,25,26,35,18,22,31,33);(1,3,4,2)(5,8)(6,12)(7,15)(9,20)(10,19)(11,22)(13,18)(14,26)(16,28)(17,29)(21,32)(23,25)(24,33)(27,31)(30,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30855);