Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13389 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2033 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,17,12,16,10,18)(7,19,23,25,22,36,30,35)(8,24,26,34,27,14,13,31)(9,21,20,15,29,33,28,32);(5,7,20,31,11,23,29,24,12,22,28,34,10,30,9,14)(6,13,21,35,18,27,32,36,16,26,33,25,17,8,15,19);(2,4)(5,8)(6,14)(7,21)(9,25)(10,26)(11,13)(12,27)(15,30)(16,24)(17,34)(18,31)(19,20)(22,33)(23,32)(28,36)(29,35)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322141, #72322145, #72322156, … |
| maximal normal subgroup of | #72322141, #72322145, #72322156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55535);