Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#453 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,11,8)(7,14,17,20)(9,13,10,18)(12,19,15,16);(1,2)(3,4)(5,7)(6,12)(8,15)(9,16)(10,19)(11,17)(13,20)(14,18);(2,4)(5,8,11,6)(7,15,17,12)(9,18,10,13)(14,19,20,16);(1,2)(3,4)(5,9,11,10)(6,13,8,18)(7,16,17,19)(12,20,15,14);(2,4)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#3018, #72312619, #72313645, … |
| maximal normal subgroup of | D4×#3018, #72312619, #72313645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480864);