Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437963 |
| 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 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,10,16,11,15,9,14)(7,17,19,27,20,29,24,12)(8,21,22,30,23,13,25,28)(18,31,33,26,34,36,32,35);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,24)(11,20)(13,26)(14,17)(15,27)(16,29)(21,35)(22,32)(23,34)(25,33)(28,31)(30,36);(2,4)(5,8)(6,13)(7,18)(9,22)(10,25)(11,23)(12,26)(14,28)(15,21)(16,30)(17,31)(19,32)(20,34)(24,33)(27,35)(29,36);(2,4)(5,9,11,10)(6,14,15,16)(7,19,20,24)(8,22,23,25)(12,17,27,29)(13,28,21,30)(18,32,34,33)(26,31,35,36);(6,15)(12,27)(13,21)(14,16)(17,29)(26,35)(28,30)(31,36)Polycyclic presentation:
2^9|a^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286439, #72286845, #72311879, … |
| maximal normal subgroup of | #72286439, #72286845, #72311879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345161);