Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
5788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(11,16)(13,14)(17,18,21,19)(20,23,24,22);(1,2)(3,5)(4,6)(7,8)(9,10,12,15)(11,14,16,13)(18,19)(22,23);(1,2,4,6)(3,7,8,5)(9,11,12,16)(10,13,15,14)(17,19,21,18)(20,23,24,22);(3,8)(5,7)(9,10,12,15)(11,14,16,13)(17,20,21,24)(18,22,19,23);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72238559, #72238560, #72238568, … |
| maximal normal subgroup of | #72238559, #72238560, #72238568, … |
| maximal quotient of | #72261954, #72300522 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9397725);