Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C23 | C22×C42 |
| socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,8,7)(13,14,18,15,17,16,19,20);(6,7)(9,10,11,12)(14,16)(15,20);(2,4)(13,15,19,14,17,20,18,16);(5,7,8,6)(13,16,18,20,17,14,19,15)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=efg,d^2=fg,g^2=i,b^a=be,c^a=cf,c^b=ci,d^b=de152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264247);