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 | |
8564 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)(7,11)(10,12)(13,14,18,16)(15,20,17,19);(1,2,3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(14,16)(19,20);(13,15,18,17)(14,19,16,20);(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,20,17,19);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,20,16,19)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72234933, #72234935 |
| maximal normal subgroup of | #72234933, #72234935 |
| maximal quotient of | #72262163 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550578);