Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×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×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1898 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)(14,18)(15,16)(19,20);(5,6,7,10,8,9,11,12)(13,14,16,20,17,19,15,18);(1,3,4,2)(14,19)(18,20);(6,9)(10,12)(14,19)(18,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=h,g^2=i,b^a=be,e^a=eh,g^a=gh,c^b=ch,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72279127, #72279203, #72280039 |
| maximal normal subgroup of | #72279127, #72279203, #72280039 |
| maximal quotient of | #72279471 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89417);