Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,9,13,11,14,10,12)(7,8,16,15,17,19,20,18);(5,6)(7,15)(8,16)(9,12)(10,13)(11,14)(17,18)(19,20);(2,4)(5,7)(6,8)(9,16)(10,20)(11,17)(12,18)(13,15)(14,19);(2,4)(5,8)(6,7)(9,18)(10,15)(11,19)(12,16)(13,20)(14,17);(5,9,11,10)(6,12,14,13)(7,16,17,20)(8,18,19,15)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,h^b=hi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213663, #72213751 |
| maximal normal subgroup of | #72213663, #72213751 |
| maximal quotient of | #72226941 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408108);