Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480460 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35512 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,15,18,19)(8,13)(9,20,16,14)(10,12)(11,17);(5,7,16,17,10,18,9,13)(6,11,20,19,12,8,14,15);(2,4)(6,12)(8,11)(14,20)(15,19);(5,8,10,11)(6,13,12,17)(7,14,18,20)(9,19,16,15)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240218, #72260917, #72260921, … |
| maximal normal subgroup of | #72240218, #72260917, #72260921, … |
| maximal quotient of | #72236774, #72306284, #72313427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401879);