Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213760 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #35348 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,12,15,10)(8,16)(9,13)(11,17)(14,19,20,18);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,19)(12,13)(16,18)(17,20);(2,4)(5,8)(6,11,13,16)(7,14)(9,17)(10,18,12,19)(15,20);(5,7)(6,12,13,10)(8,14)(9,15)(11,19,16,18)(17,20)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225631, #72225637, #72225642, … |
| maximal normal subgroup of | #72225631, #72225637, #72225642, … |
| maximal quotient of | #72251455, #72265648 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290937);