Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #386 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,10,12,8,7,9,11)(14,15,19,20);(9,10)(11,12)(13,14,17,15,18,19,16,20);(1,2,3,4)(5,7,8,6)(9,11,10,12)(13,15,16,14,18,20,17,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efh,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=ehi,f^c=fhi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239507 |
| maximal normal subgroup of | #72239507 |
| maximal quotient of | #72239512, #72239514 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17374);