Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3522 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,4,3)(5,6,10,9)(7,12,8,11)(13,14)(15,19)(16,18)(17,20);(5,7,10,8)(6,11,9,12)(13,14,16,18)(15,17,20,19);(1,3,4,2)(5,7,10,8)(6,11,9,12)(13,14)(15,17)(16,18)(19,20);(5,7,10,8)(6,11,9,12)(13,15,16,20)(14,17,18,19);(5,8,10,7)(6,11,9,12);(5,9,10,6)(7,12,8,11)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534527);