Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
837 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)(5,6,12,15,11,16,10,8)(7,17,27,13,20,21,19,30)(9,23,28,22,26,14,25,32)(18,33,35,36,34,31,24,29);(5,7,10,27,11,20,12,19)(6,13,8,21,16,30,15,17)(9,24,25,18,26,35,28,34)(14,31,22,29,23,33,32,36);(5,8,10,16,11,15,12,6)(7,13,27,21,20,30,19,17)(9,22,25,23,26,32,28,14)(18,29,35,33,34,36,24,31);(2,4)(5,9,10,25,11,26,12,28)(6,14,8,22,16,23,15,32)(7,18,19,24,20,34,27,35)(13,29,17,31,30,36,21,33)Polycyclic presentation:
2^9|a^2=eh,b^2=ef,c^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322821 |
| maximal normal subgroup of | #72322821 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357790);