Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
11886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(2,4)(5,6,7,8);(5,6)(7,8)(10,13)(14,16);(1,2,3,4)(9,10)(11,14)(12,13)(15,16);(11,15)(14,16);(9,11)(10,14)(12,15)(13,16);(9,10)(11,14)(12,13)(15,16)Polycyclic presentation:
2^9|a^2=g,c^2=h,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72233759, #72233769, #72234460, … |
| maximal normal subgroup of | #72233759, #72233769, #72234460, … |
| maximal quotient of | #72261938, #72262129, #72262157 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538529);