Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35222 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,13,10,14,9,12)(7,11,15,20,17,18,16,19);(2,4)(5,7,8,15,10,17,9,16)(6,11,13,20,14,18,12,19);(6,12)(7,15)(8,9)(11,18)(13,14)(16,17);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214076, #72215972, #72216029, … |
| maximal normal subgroup of | #72214076, #72215972, #72216029, … |
| maximal quotient of | C2×#364424, #72236752, #72239574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271620);