Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×D42 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,15,17,16,18,19,20);(1,2)(3,4)(5,7,9,8)(6,10,12,11);(6,11)(7,8)(10,12);(2,4)(6,12)(10,11)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ch,d^b=de,f^c=fh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126999);