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 | |
12228 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)(6,8)(9,10,15,16)(11,14,13,12);(5,6)(7,8)(9,11,15,13)(10,12,16,14);(2,3)(9,12,15,14)(10,13,16,11);(1,3,4,2)(9,11,15,13)(10,12,16,14);(9,13,15,11)(10,12,16,14);(9,14,15,12)(10,13,16,11)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72234475, #72234479, #72236840, … |
| maximal normal subgroup of | #72234475, #72234479, #72236840, … |
| maximal quotient of | #72261995, #72262008, #72262068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7606276);