Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(13,14)(15,19)(16,18)(17,20);(1,2)(3,4)(5,6,9,8)(7,10,12,11);(6,8)(10,11)(13,14,16,18)(15,17,20,19);(5,7,9,12)(6,10,8,11)(13,14)(15,17)(16,18)(19,20);(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,20)(18,19);(5,8,9,6)(7,11,12,10)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72233742, #72233743, #72233752, … |
| maximal normal subgroup of | #72233742, #72233743, #72233752, … |
| maximal quotient of | #72260971, #72261165, #72261346 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7691211);