Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437931 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,11,14)(7,9,13,10)(8,12,17,20)(15,16,19,18);(5,7,9,14,11,13,10,6)(8,12,16,15,17,20,18,19);(1,2,3,4)(5,8)(6,12)(7,15)(9,16)(10,18)(11,17)(13,19)(14,20);(5,8)(6,12)(7,15)(9,16)(10,18)(11,17)(13,19)(14,20);(5,9,11,10)(6,13,14,7)(8,16,17,18)(12,19,20,15)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72227253, #72227280, D4×#3021, … |
| maximal normal subgroup of | #72227253, #72227280, D4×#3021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350511);