Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
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 | C2×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,12,9,11,7,10)(13,14,17,19,16,18,20,15);(1,2,3,4)(6,10)(7,8)(11,12)(14,18)(15,19);(5,7,9,8)(6,10,11,12)(13,15,20,18,16,19,17,14);(6,11)(10,12)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,b^a=be,d^a=dh,c^b=ci,e^b=eh,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72279535, #72279582, #72279630, … |
| maximal normal subgroup of | #72279535, #72279582, #72279630, … |
| maximal quotient of | #72342630 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90645);