Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,8,10,9,11,7,12);(6,10)(7,8)(11,12)(13,14,15,18,16,17,19,20);(1,3,4,2)(6,11)(10,12);(14,17)(18,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=h,g^2=i,b^a=be,c^a=ch,d^b=di,e^b=eh,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | M16×#442, #72279607, #72279671, … |
| maximal normal subgroup of | M16×#442, #72279607, #72279671, … |
| maximal quotient of | #72309649, #72342620 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90599);