Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2023 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,12,17,28,35,10,18,25,29,20,36,11,16)(7,19,23,34,24,30,26,13,22,31,21,32,27,33,8,14);(1,2)(3,4)(5,7,10,22)(6,13)(8,25,26,9)(11,23,28,21)(12,27,20,24)(14,18)(15,30)(16,31)(17,34)(19,35)(29,33)(32,36);(2,4)(5,8,9,22,12,23,28,27,10,26,25,7,20,21,11,24)(6,14,15,31,17,34,35,33,18,13,29,19,36,32,16,30)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dh,f^b=fh,g^b=ghi,h^b=hi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250251, #72250266, #72250316, … |
| maximal normal subgroup of | #72250251, #72250266, #72250316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56241);