Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442584 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,11)(7,14)(8,9)(12,19)(13,17)(15,16);(1,2)(3,4)(5,6)(7,12)(8,17)(9,11)(10,13)(14,18)(15,19)(16,20);(2,4)(5,7,9,14,10,15,8,16)(6,12,11,18,13,19,17,20);(2,4);(7,15)(12,19)(14,16)(18,20)Polycyclic presentation:
2^9|c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213663, #72213818, #72213852, … |
| maximal normal subgroup of | #72213663, #72213818, #72213852, … |
| maximal quotient of | #72226922, C2×#6442584, #72239140, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349780);