Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437969 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,8,11,9,12,16,15)(7,10,14,18,13,17,19,20);(1,2)(3,4)(5,6)(7,10)(8,15)(9,12)(11,16)(13,17)(14,20)(18,19);(2,4)(8,16)(11,15)(14,19)(18,20);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19);(7,13)(10,17)(14,19)(18,20)Polycyclic presentation:
2^9|a^2=f,f^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213210, #72213250, #72213264, … |
| maximal normal subgroup of | #72213210, #72213250, #72213264, … |
| maximal quotient of | #72260614, C2×#6437969 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345165);