Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,10)(7,13)(8,15)(11,18)(12,17)(16,19);(5,6)(7,11)(8,10,15,17)(9,12)(13,19)(14,16,20,18);(1,3,4,2)(7,13)(11,19)(14,20)(16,18);(5,7,9,13)(6,11,12,19)(8,14,15,20)(10,16,17,18)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=i,b^a=be,d^a=di,e^a=ei,g^a=ghi,e^b=eh,h^b=hi,d^c=di,g^e=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72310464, C4×#33359, #72310482, … |
| maximal normal subgroup of | #72310464, C4×#33359, #72310482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400231);