Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,16,11,14,10,27)(7,17,19,31,20,32,26,13)(8,21,23,33,25,35,24,15)(12,22,29,34,28,36,30,18);(5,7)(6,12)(8,22)(9,19)(10,26)(11,20)(13,15)(14,28)(16,29)(17,21)(18,24)(23,34)(25,36)(27,30)(31,33)(32,35);(2,4)(5,7)(6,13)(8,18)(9,26)(10,19)(11,20)(12,15)(14,31)(16,32)(17,27)(21,30)(22,24)(23,36)(25,34)(28,33)(29,35);(6,14)(12,28)(13,31)(15,33)(16,27)(17,32)(21,35)(29,30)Polycyclic presentation:
2^9|a^2=g,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,h^b=hi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231513, #72231532, #72231549, … |
| maximal normal subgroup of | #72231513, #72231532, #72231549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400747);