Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,16,18,19)(8,17,20,13)(9,12)(10,14)(11,15);(2,4)(6,12)(8,19)(9,10)(13,17)(14,15)(16,20);(1,2,3,4)(5,7,10,17,11,18,9,13)(6,8,12,19,15,20,14,16);(5,8)(6,13)(7,12)(9,16)(10,19)(11,20)(14,18)(15,17)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237184, #72265640, #72265644, … |
| maximal normal subgroup of | #72237184, #72265640, #72265644, … |
| maximal quotient of | #72239111, #72239112, #72286796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293105);