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 | |
8115 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)(5,6)(7,12)(8,11)(9,10);(1,2)(3,4)(7,8)(11,12)(13,14)(15,17)(16,18)(19,20);(5,7,10,8)(6,11,9,12)(13,14,16,18)(15,17,19,20);(7,8)(11,12)(15,19)(17,20);(5,8,10,7)(6,11,9,12)(13,15,16,19)(14,17,18,20);(5,9,10,6)(7,11,8,12)(13,14)(15,17)(16,18)(19,20)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72262456, #72262499, #72262551, … |
| maximal normal subgroup of | #72262456, #72262499, #72262551, … |
| maximal quotient of | #72300307, #72300370, #72300415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7920332);