Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437478 |
| 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×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4733 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)(7,8)(9,13)(10,14)(11,15)(12,16)(17,19)(18,20);(5,7)(6,12)(8,14)(9,16)(10,20)(11,17)(13,19)(15,18);(1,2,3,4)(5,6)(7,8)(9,14)(10,13)(11,15)(12,20)(16,18)(17,19);(5,8)(6,7)(9,12)(10,18)(11,19)(13,16)(14,20)(15,17);(5,9,11,10)(6,13,15,14)(7,16,17,20)(8,12,19,18)Polycyclic presentation:
2^9|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,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286288, #72341724, #72341782, … |
| maximal normal subgroup of | #72286288, #72341724, #72341782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349810);