Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437608 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 | |
6167 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)(7,15)(8,13)(9,11)(10,14)(12,17)(16,20)(18,19);(2,4)(6,11)(7,16)(8,9)(13,14)(15,19)(17,18);(5,7)(6,12)(8,16)(9,17)(10,18)(11,19)(13,15)(14,20);(5,6)(7,12)(8,11)(9,13)(10,14)(15,17)(16,19)(18,20);(5,8,10,9)(6,11,14,13)(7,16,18,17)(12,19,20,15)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213527, #72213818, #72226293, … |
| maximal normal subgroup of | #72213527, #72213818, #72226293, … |
| maximal quotient of | #72226259, #72310311, #72310313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344806);