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 | |
8453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2)(3,4)(5,6,10,9)(7,12,11,8)(14,17)(18,20);(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20);(2,4)(5,8)(6,11)(7,9)(10,12);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11);(5,9,10,6)(7,12,11,8)(13,15,16,19)(14,18,17,20);(5,7)(6,12)(8,9)(10,11)Polycyclic presentation:
2^9|a^2=i,e^2=g,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 | #72219286, #72232192, #72234902, … |
| maximal normal subgroup of | #72219286, #72232192, #72234902, … |
| maximal quotient of | #72219259, #72219260, #72232262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7603189);