Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3514 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)(7,11)(8,10)(9,12)(13,14)(15,17,19,20)(16,18);(5,7,8,12)(6,9,10,11);(2,4)(7,12)(9,11)(13,15)(14,17,18,20)(16,19);(6,10)(9,11)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,h^a=hi,c^b=ce,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72231639, #72231659, #72231663, … |
| maximal normal subgroup of | #72231639, #72231659, #72231663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263196);