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 | |
8074 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)(13,14)(15,20)(16,18)(17,19);(1,2)(3,4)(5,6,9,7)(8,11,12,10)(13,15,17,16)(14,18,19,20);(6,7)(10,11)(14,19)(18,20);(13,14)(15,18)(16,20)(17,19);(5,7,9,6)(8,11,12,10)(13,16,17,15)(14,20,19,18);(5,8,9,12)(6,10,7,11)Polycyclic presentation:
2^9|b^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72238150, #72238161, #72238166, … |
| maximal normal subgroup of | #72238150, #72238161, #72238166, … |
| maximal quotient of | #72297286, #72297290, #72297455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7639786);