Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33618 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,12,8,11,22)(7,16,20,33,18,32,28,34)(9,23,25,21,27,36,26,14)(13,29,30,19,17,35,31,24);(5,7,9,24,11,20,26,29,12,18,27,19,10,28,25,35)(6,13,21,16,15,31,36,34,8,17,14,32,22,30,23,33);(2,4)(5,8)(6,12)(7,17)(9,14)(10,15)(11,22)(13,18)(16,19)(20,30)(21,27)(23,26)(24,32)(25,36)(28,31)(29,33)(34,35);(7,18)(13,17)(16,32)(19,24)(20,28)(29,35)(30,31)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321342, #72321394, #72321415, … |
| maximal normal subgroup of | #72321342, #72321394, #72321415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401855);