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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(14,18)(15,16);(5,7)(6,9,10,11)(8,12)(13,14,16,20,17,18,15,19);(1,3,4,2)(14,18)(19,20);(13,15,17,16)(14,19,18,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,g^2=i,b^a=be,d^a=di,g^a=gh,c^b=ci,e^b=eh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72309174, #72309181, #72309242, … |
| maximal normal subgroup of | #72309174, #72309181, #72309242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89185);