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 | |
12170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2,4,3)(5,6,7,8)(10,14)(12,15);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16);(2,3)(10,14)(11,16);(1,3,4,2);(9,11)(10,15)(12,14)(13,16);(9,12,13,15)(10,11,14,16)Polycyclic presentation:
2^9|a^2=ghi,d^2=h,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72233752, #72233754, #72233755, … |
| maximal normal subgroup of | #72233752, #72233754, #72233755, … |
| maximal quotient of | #72257084, #72257103, #72257911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7587456);